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Journal of the Association of Arab Universities for Basic and Applied Sciences (2016) 21,1–9

University of Bahrain Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com

ORIGINAL ARTICLE Facile synthesis and antimicrobial activity of a novel series of 7,8-dihydro-2-(2-oxo-2H- chromen-3-yl)-5-aryl-cyclopenta[b] pyrano-pyrimidine-4,6-5H-dione derivatives catalyzed by reusable silica-bonded N-propyl diethylenetriamine sulfamic acid

Prasanna Nithiya Sudhan, Syed Sheik Mansoor *

Research Department of Chemistry, Bioactive Organic Molecule Synthetic Unit, C. Abdul Hakeem College, Melvisharam 632 509, Tamil Nadu, India

Received 5 June 2014; revised 28 November 2014; accepted 22 December 2014 Available online 21 February 2015

KEYWORDS Abstract An efficient method for the synthesis of a novel series of cyclopenta[b]pyrano Cyclopenta[b]pyrano pyrimidinone derivatives with silica-bonded N-propyl diethylenetriamine sulfamic acid (SBPDSA) pyrimidinones; as catalyst has been achieved by the condensation of 2-amino-4-phenyl-5-oxo-4,5,6,7-tetrahydrocy- Silica-bonded N-propyl clopenta[b]pyran-3-carbonitrile derivatives and coumarin-3-carboxylic acid under solvent free con- diethylenetriamine sulfamic ditions. Antimicrobial studies showed all the target compounds processing good antibacterial and acid; antifungal activities. Coumarin-3-carboxylic acid; ª 2014 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the Antimicrobial activity CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction pharmacological properties such as anti-inflammatory (Witaicenis et al., 2014), antitumor (Avin et al., 2014), anticancer Coumarins (2-oxo-2H-chromenes) are an old class of com- (Jashari et al., 2014; Zhang et al., 2014), acetylcholinesterase pounds, also known as benzopyranes, comprising a large class (AChE) and butyrylcholinesterase (BuChE) inhibitors of cinnamic acid-derived phenolic compounds found in fungi, (Asadipour et al., 2013), anti-proliferative (Zhao et al., 2014), bacteria and plants, particularly in edible plants from different antibacterial, antifungal and antioxidant (Renuka and Kumar, botanical families. Coumarins and their derivatives have attract- 2013), anti-osteoporotic (Sashidhara et al., 2013) and anti-tuber- ed intense interest in recent years because of their diverse culosis activities (Kawate et al., 2013). A considerable effort has been made for the synthesis of heterocyclic compounds containing coumarin moiety due to * Corresponding author. their wide pharmaceutical importance (Banothu and E-mail address: [email protected] (S. Sheik Mansoor). Bavanthula, 2012; Ghosh and Das, 2012; Augustine et al., Peer review under responsibility of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2014.12.001 1815-3852 ª 2014 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2 P.N. Sudhan, S. Sheik Mansoor

2012; Khoobi et al., 2011; Khan et al., 2011, 2012; Khurana obtained as a white powder. The content of S obtained from and Kumar, 2009). However, these procedures are not entirely elemental analysis showed that typically a loading of satisfactory and suffer from long reaction time or tedious work 0.99 mmol/g H+ was obtained (Rahi et al., 2012). up. Hence, a method using a nonmetallic catalyst is desirable. Therefore, the introduction of new and efficient methods for 2.3. General procedure to synthesis of 2-amino-4-phenyl-5-oxo- this reaction is still necessary. Toward this goal, we were 4,5,6,7-tetrahydrocyclopenta[b] pyran-3-carbonitrile derivatives prompted to explore new methods for the synthesis of hetero- using Alum (KAl(SO4)2.12H2O) (10 mol%) as catalyst cyclic compounds containing coumarin moiety. Silica-bonded N-propyl diethylenetriamine sulfamic acid has A mixture of aldehydes 1 (1 mmol), malononitrile 2 been reported as a novel catalyst for chemoselective synthesis of (1 mmol), cyclopentane-1,3-dione 3 (1 mmol), and powdered 1,1-diacetates (Sefat et al., 2011), and synthesis of a-aminoni- Alum (KAl(SO4)2Æ12H2O) (10 mol%), under solvent-free triles (Rahi et al., 2012). However, to the best of our knowledge, conditions was stirred at 70 C for appropriate time there are no examples on the use of SBPDSA as catalyst for the (Scheme 1). The progress of the reaction was monitored synthesis of 7,8-dihydro-2-(2-oxo-2H-chromen-3-yl)-5-aryl-cy- by TLC. After completion, the reaction was allowed to cool, clopenta[b]pyrano-pyrimidine-4,6-5H-dione derivatives. ethanol (20 mL) was added and the catalyst was recovered Considering the potential of developing new routes to the syn- to use subsequently by filtration. Concentration of the fil- thesis of heterocyclic compounds containing coumarin moiety due trate and recrystallization of the solid residue from hot etha- to their wide pharmaceutical importance (Ghashang et al., 2013, nol afforded the crystals of 2-amino-4-phenyl-5-oxo-4,5,6,7- 2014a,b), we now describe the synthesis of 7,8-dihydro-2-(2-oxo- tetrahydrocyclopenta[b]pyran-3-carbonitriles in high yield. 2H-chromen-3-yl)-5-phenyl-cyclopenta[b]pyrano-pyrimidine-4,6- The recovered catalyst can be washed consequently with 5H-dione derivatives by the condensation of 2-amino-4-phenyl-5- an aliquot of fresh CH2Cl2 (2 · 10 mL), water and then ace- oxo-4,5,6,7-tetrahydrocyclopenta[b]pyran-3-carbonitrile deriva- tone. After drying, it can be reused without noticeable loss tives and coumarin-3-carboxylic acid using SBPDSA as an efficient of reactivity. Compounds 4a–j were identified by IR, 1H novel catalyst. NMR, 13C NMR, mass and elemental analysis. The synthesis of 2-amino-4-phenyl-5-oxo-4,5,6,7-tetrahy- drocyclopenta[b]pyran-3-carbonitrile derivatives was achieved 2.4. Spectral data for the synthesized compounds (4a–j) by the condensation of aldehydes 1, malononitrile 2 and cyclopentane-1,3-dione 3, using Alum (KAl(SO4)2Æ12H2O) as 2.4.1. 2-Amino-4-phenyl-5-oxo-4,5,6,7- catalyst under solvent-free conditions (Scheme 1). tetrahydrocyclopenta[b]pyran-3-carbonitrile (4a) IR (KBr, cmÀ1): 3392, 3322, 3220, 2201, 1682, 1604, 1512, 2. Experimental 1 1356, 1061, 683; H NMR (500 MHz, DMSO-d6) d: 2.22– 2.34 (m, 2H, CH2), 2.54–2.65 (m, 2H, CH2), 4.24 (s, 1H, 2.1. Apparatus and analysis CH), 6.66 (s, 2H, NH2), 7.18 (d, J = 7.2 Hz, 2H, ArH), 7.38 13 (m, 3H, ArH) ppm; C NMR (125 MHz, DMSO-d6) d: Chemicals were purchased from Merck, Fluka and Aldrich 26.0, 35.2, 39.0, 49.2, 58.3, 112.0, 119.4, 126.3, 127.3, 129.0, + Chemical Companies. All yields refer to isolated products 144.0, 157.9, 163.0, 195.6 ppm; MS(ESI): m/z 253 (M+H) ; 1 13 unless otherwise stated. H NMR (500 MHz) and C NMR Anal. Calcd for C15H12N2O2: C, 71.42; H, 4.76; N, 11.11%. (125 MHz) spectra were obtained using a Bruker DRX-500 Found: C, 71.36; H, 4.71; N, 11.10%. Avance spectrometer at ambient temperature, using TMS as internal standard. FT-IR spectra were recorded as KBr pellets 2.4.2. 2-Amino-4-(4-methylphenyl)-5-oxo-4,5,6,7- on a Shimadzu spectrometer. Mass spectra were determined on tetrahydrocyclopenta[b]pyran-3-carbonitrile (4b) a Varian-Saturn 2000 GC/MS instrument. Elemental analysis IR (KBr, cmÀ1): 3396, 3320, 3226, 2195, 1665, 1608, 1514, 1 was measured by means of a Perkin Elmer 2400 CHN elemen- 1364, 1036, 798; H NMR (500 MHz, DMSO-d6) d: 2.18 (s, tal analyzer flowchart. 3H, CH3), 2.20–2.34 (m, 2H, CH2), 2.55–2.67 (m, 2H, CH2), 4.19 (s, 1H, CH), 6.74 (s, 2H, NH2), 7.22 (d, J = 7.4 Hz, 2.2. Preparation of silica-bonded N-propyl diethylenetriamine 2H, ArH), 7.40 (d, J = 7.4 Hz, 2H, ArH) ppm; 13C NMR sulfamic acid (SBPDSA) (125 MHz, DMSO-d6) d: 26.7, 35.3, 38.7, 49.4, 58.0, 112.6, 119.2, 126.4, 127.3, 129.3, 143.7, 158.3, 162.9, 196.0 ppm; + The catalyst was prepared as per the previously reported MS(ESI): m/z 267 (M+H) ; Anal. Calcd for C16H14N2O2: method (Sefat et al., 2011). The catalyst SBPDSA was C, 72.18; H, 5.26; N, 10.52%. Found: C, 72.22; H, 5.25; N, 10.50%.

R1

R1 4a H 2.4.3. 2-Amino-4-(4-nitrophenyl)-5-oxo-4,5,6,7- R 1 4b 4-CH3 tetrahydrocyclopenta[b]pyran-3-carbonitrile (4c) Alum 4c 4-NO2 10 mol% O 4d 3-Br À1 CHO CN IR (KBr, cm ): 3400, 3324, 3228, 2206, 1673, 1606, 1522, O 4e 4-Cl 1a-j 70 oC 4f 3-OH 1 1355, 1073, 836; H NMR (500 MHz, DMSO-d6) d: 2.18– Solvent-free O NH 4g 4-OH + NC CN 2 4h 4-N(CH ) 2.27 (m, 2H, CH ), 2.53–2.61 (m, 2H, CH ), 4.22 (s, 1H, 2 4a- j 3 2 2 2 4i 3-CH 3 3 CH), 6.80 (s, 2H, NH ), 7.26 (d, J = 7.0 Hz, 2H, ArH), 7.45 O 4j 3-OCH3 2 (d, J = 7.1 Hz, 2H, ArH) ppm; 13C NMR (125 MHz, Scheme 1 Preparation of various 2-amino-4-phenyl-5-oxo- DMSO-d6) d: 26.4, 35.4, 39.4, 49.1, 57.9, 113.0, 118.9, 125.9, 4,5,6,7-tetrahydrocyclopenta [b]pyran-3-carbonitrile derivatives. 127.3, 128.9, 143.8, 158.5, 163.8, 196.2 ppm; MS(ESI): m/z Synthesis of cyclopenta[b]pyrano pyrimidinone derivatives 3

+ 298 (M+H) ; Anal. Calcd for C15H11N3O4: C, 60.60; H, 3.70; 59.2, 112.9, 118.7, 126.3, 127.3, 129.3, 144.5, 158.0, 163.4, N, 14.14%. Found: C, 60.50; H, 3.66; N, 14.10%. 196.1 ppm; MS(ESI): m/z 296 (M+H)+; Anal. Calcd for C17H17N3O2: C, 69.15; H, 5.76; N, 14.23%. Found: C, 69.07; 2.4.4. 2-Amino-4-(3-bromophenyl)-5-oxo-4,5,6,7-tetrahydrocyc H, 5.71; N, 14.19%. lopenta[b]pyran-3-carbonitrile (4d) IR (KBr, cmÀ1): 3403, 3318, 3212, 2207, 1678, 1611, 1510, 2.4.9. 2-Amino-4-(3-methylphenyl)-5-oxo-4,5,6,7-tetrahydroc 1 yclopenta[b]pyran-3-carbonitrile (4i) 1360, 1074, 845; H NMR (500 MHz, DMSO-d6) d: 2.09– À1 2.18 (m, 2H, CH2), 2.44–2.53 (m, 2H, CH2), 4.26 (s, 1H, IR (KBr, cm ): 3388, 3322, 3222, 2204, 1675, 1607, 1514, 13 1 CH), 6.85 (s, 2H, NH2), 7.31–7.44 (m, 4H, ArH) ppm; C 1362, 1077, 790; H NMR (500 MHz, DMSO-d6) d: 2.12– NMR (125 MHz, DMSO-d6) d: 27.0, 36.0, 38.7, 49.7, 58.6, 2.19 (m, 2H, CH2), 2.24 (s, 3H, CH3), 2.42–2.53 (m, 2H, 113.2, 118.8, 126.3, 127.3, 128.7, 144.4, 158.8, 164.0, CH2), 4.12 (s, 1H, CH), 6.76 (s, 2H, NH2), 7.30–7.45 (m, + 13 195.9 ppm; MS(ESI): m/z 331.9 (M+H) ; Anal. Calcd for 4H, ArH) ppm; C NMR (125 MHz, DMSO-d6) d: 26.4, C15H11BrN2O4: C, 54.40; H, 3.32; N, 8.46%. Found: C, 35.7, 38.3, 50.2, 58.3, 112.9, 118.9, 125.8, 127.3, 128.8, 143.8, 54.42; H, 3.30; N, 8.44%. 157.9, 163.6, 196.0 ppm; MS(ESI): m/z 267 (M+H)+; Anal. Calcd for C16H14N2O2: C, 72.18; H, 5.26; N, 10.52%. Found: 2.4.5. 2-Amino-4-(4-chlorophenyl)-5-oxo-4,5,6,7-tetrahydrocyc C, 72.12; H, 5.22; N, 10.48%. lopenta[b]pyran-3-carbonitrile (4e) IR (KBr, cmÀ1): 3394, 3331, 3214, 2208, 1669, 1600, 1515, 2.4.10. 2-Amino-4-(3-methoxyphenyl)-5-oxo-4,5,6,7-tetrahydro 1 cyclopenta[b]pyran-3-carbonitrile (4j) 1362, 1071, 844; H NMR (500 MHz, DMSO-d6) d: 2.16– À1 2.28 (m, 2H, CH2), 2.51–2.63 (m, 2H, CH2), 4.16 (s, 1H, IR (KBr, cm ): 3405, 3326, 3216, 2195, 1681, 1597, 1523, 1 CH), 6.74 (s, 2H, NH2), 7.25 (d, J = 7.2 Hz, 2H, ArH), 7.38 1355, 1031, 849, 762; H NMR (500 MHz, DMSO-d6) d: 13 (d, J = 7.2 Hz, 2H, ArH) ppm; C NMR (125 MHz, 2.13–2.24 (m, 2H, CH2), 2.38–2.50 (m, 2H, CH2), 3.58 (s, DMSO-d6) d: 26.8, 36.1, 39.4, 49.4, 58.6, 112.8, 119.4, 126.5, 3H, OCH3), 4.17 (s, 1H, CH), 6.69 (s, 2H, NH2), 7.19–7.33 13 127.3, 128.4, 144.0, 158.6, 163.6, 196.1 ppm; MS(ESI): m/z (m, 4H, ArH) ppm; C NMR (125 MHz, DMSO-d6) d: + 287 (M+H) ; Anal. Calcd for C15H11ClN2O2: C, 62.84; H, 27.4, 36.4, 38.7, 49.7, 59.3, 113.5, 118.2, 125.6, 127.3, 129.4, 3.84; N, 9.77%. Found: C, 62.77; H, 3.80; N, 9.75%. 144.0, 158.2, 163.7, 195.8 ppm; MS(ESI): m/z 283 (M+H)+; Anal. Calcd for C16H14N2O3: C, 68.08; H, 4.96; N, 9.92%. 2.4.6. 2-Amino-4-(3-hydroxyphenyl)-5-oxo-4,5,6,7-tetrahydroc Found: C, 68.00; H, 4.94; N, 9.88%. yclopenta[b]pyran-3-carbonitrile (4f) IR (KBr, cmÀ1): 3432, 3390, 3329, 3219, 2196, 1680, 1598, 2.5. General procedure to synthesis of 7,8-dihydro-2-(2-oxo-2H- 1 chromen-3-yl)-5-phenyl-cyclopenta[b]pyrano-pyrimidine-4,6- 1524, 1351, 1066, 856; H NMR (500 MHz, DMSO-d6) d: 5H-dione derivatives using SBPDSA as catalyst 2.11–2.23 (m, 2H, CH2), 2.55–2.66 (m, 2H, CH2), 4.21 (s, 1H, CH), 6.80 (s, 2H, NH2), 7.34–7.49 (m, 4H, ArH), 9.48 13 (s, 1H, OH) ppm; C NMR (125 MHz, DMSO-d6) d: 27.2, A mixture of 2-amino-4-phenyl-5-oxo-4,5,6,7-tetrahydrocy- 35.7, 38.6, 50.0, 59.0, 113.4, 118.8, 125.7, 127.3, 129.0, 143.6, clopenta[b]pyran-3-carbonitrile 4a–j (1 mmol), coumarin-3- 158.4, 163.8, 195.7 ppm; MS(ESI): m/z 269 (M+H)+; Anal. carboxylic acid 5 (1 mmol) and SBPDSA (0.051 g/5 mol%) Calcd for C15H12N2O3: C, 67.16; H, 4.48; N, 10.45%. Found: was heated at 80 C for about 4–6 h (Scheme 2). After comple- C, 67.10; H, 4.44; N, 10.40%. tion of the reaction (TLC), 2 mL of water was added, and the reaction mixture was stirred at room temperature for 20 min. 2.4.7. 2-Amino-4-(4-hydroxyphenyl)-5-oxo-4,5,6,7-tetrahydroc The resulting precipitate was filtered. The crude product was yclopenta[b]pyran-3-carbonitrile (4g) purified by column chromatography (n-hexane/ethyl acetate, IR (KBr, cmÀ1): 3466, 3401, 3333, 3289, 2194, 1677, 1596, 80:20) to provide the pure products. Compounds 6a–j were 1 identified by IR, 1H NMR, 13C NMR, mass and elemental 1525, 1358, 1074, 844; H NMR (500 MHz, DMSO-d6) d: analysis. 2.06–2.14 (m, 2H, CH2), 2.43–2.56 (m, 2H, CH2), 4.27 (s, 1H, CH), 6.82 (s, 2H, NH2), 7.19 (d, J = 7.1 Hz, 2H, ArH), 7.35 (d, J = 7.2 Hz, 2H, ArH), 9.56 (s, 1H, OH) ppm; 13C 2.6. Spectral data for the synthesized compounds (6a–j) NMR (125 MHz, DMSO-d6) d: 26.5, 36.1, 39.3, 50.2, 58.4, 112.7, 118.7, 126.7, 127.3, 129.2, 144.2, 158.4, 163.0, 2.6.1. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-phenyl- 196.2 ppm; MS(ESI): m/z 269 (M+H)+; Anal. Calcd for cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6a) C15H12N2O3: C, 67.16; H, 4.48; N, 10.45%. Found: C, 67.06; IR (KBr, cmÀ1): 3411, 1713, 1659, 1620, 1599, 1211, 1066, 701; H, 4.45; N, 10.43%. 1 H NMR (500 MHz, DMSO-d6) d: 2.04–2.15 (m, 2H, CH2), 2.44–2.56 (m, 2H, CH2), 4.44 (s, 1H, CH), 7.12 (d, 2.4.8. 2-Amino-4-(4-N,N-dimethylaminophenyl)-5-oxo-4,5,6,7- J = 7.2 Hz, 2H, ArH), 7.44–7.51 (m, 3H, ArH), 7.66–7.78 tetrahydrocyclopenta[b]pyran-3-carbonitrile (4h) (m, 4H, ArH), 8.46 (s, 1H, Coumarin), 9.48 (s, 1H, NH) À1 13 IR (KBr, cm ): 3398, 3320, 3210, 2199, 1672, 1604, 1516, ppm; C NMR (125 MHz, DMSO-d6) d: 16.5, 20.3, 26.7, 1 1364, 1033, 855; H NMR (500 MHz, DMSO-d6) d: 2.20– 35.9, 37.2, 100.0, 113.2, 115.9, 118.2, 118.9, 124.9, 129.3, 2.32 (m, 2H, CH2), 2.50–2.61 (m, 2H, CH2), 2.72 (s, 6H, 131.0, 134.0, 136.5, 152.5, 154.6, 156.9, 164.7, 195.4 ppm; + N(CH3)2), 4.29 (s, 1H, CH), 6.78 (s, 2H, NH2), 7.26 (d, MS(ESI): m/z 425 (M+H) ; Anal. Calcd for C25H16N2O5: J = 7.2 Hz, 2H, ArH), 7.41 (d, J = 7.2 Hz, 2H, ArH) ppm; C, 70.75; H, 3.77; N, 6.60%. Found: C, 70.70; H, 3.75; N, 13 C NMR (125 MHz, DMSO-d6) d: 27.2, 37.3, 38.6, 49.6, 6.58%. 4 P.N. Sudhan, S. Sheik Mansoor

R1 R1 O O R1 O SBPDSA 6a H HOOC CN 0.051 g (5 mol%) NH 6b 4-CH3 + 6c 4-NO2 Solvent-free O N 6d 3-Br O NH O O 2 80 oC 6e 4-Cl 4a-j 5 O O 6f 3-OH 6a- j 6g 4-OH 6 h 4-N(CH3)2 6i 3-CH3 SO3H SO3H 6j 3-OCH3 O N N O N H SiO2 Si O SO3H

(SBPDSA)

Scheme 2 Preparation of 7,8-dihydro-2-(2-oxo-2H-chromen-3-yl)-5-aryl-cyclopenta[b] pyrano-pyrimidine-4,6-5H-diones.

2.6.2. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(4-methy J = 7.2 Hz, 2H, ArH), 7.28 (d, J = 7.2 Hz, 2H, ArH), 7.73– lphenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6b) 7.86 (m, 4H, ArH), 8.38 (s, 1H, Coumarin), 9.50 (s, 1H, 13 IR (KBr, cmÀ1): 3406, 1700, 1660, 1622, 1603, 1205, 1044, 788; NH) ppm; C NMR (125 MHz, DMSO-d6) d: 16.0, 20.0, 1 26.2, 35.9, 37.7, 101.3, 113.9, 115.7, 118.2, 118.8, 124.8, H NMR (500 MHz, DMSO-d6) d: 2.11–2.21 (m, 2H, CH2), 2.28 (s, 3H, CH ), 2.52–2.64 (m, 2H, CH ), 4.50 (s, 1H, CH), 129.3, 130.7, 133.8, 137.0, 153.4, 154.1, 156.7, 164.4, 3 2 + 7.22 (d, J = 7.2 Hz, 2H, ArH), 7.51 (d, J = 7.2 Hz, 2H, 195.4 ppm; MS(ESI): m/z 459.45 (M+H) ; Anal. Calcd for ArH), 7.60–7.77 (m, 4H, ArH), 8.40 (s, 1H, Coumarin), 9.40 C25H15ClN2O5: C, 65.43; H, 3.27; N, 6.11%. Found: C, 13 65.40; H, 3.25; N, 6.10%. (s, 1H, NH) ppm; C NMR (125 MHz, DMSO-d6) d: 16.6, 20.4, 26.5, 35.7, 36.9, 101.0, 113.8, 115.7, 117.9, 118.4, 124.3, 129.3, 130.5, 134.2, 137.0, 153.0, 154.0, 157.1, 163.9, 2.6.6. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(3-hydro 194.9 ppm; MS(ESI): m/z 439 (M+H)+; Anal. Calcd for xyphenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6f) À1 C26H18N2O5: C, 71.23; H, 4.11; N, 6.39%. Found: C, 71.13; IR (KBr, cm ): 3455, 3403, 1698, 1669, 1622, 1596, 1210, 1 H, 4.09; N, 6.36%. 1043, 846; H NMR (500 MHz, DMSO-d6) d: 2.02–2.12 (m, 2H, CH2), 2.33–2.44 (m, 2H, CH2), 4.55 (s, 1H, CH), 7.24– 2.6.3. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(4-nitro 7.46 (m, 4H, ArH), 7.77–7.88 (m, 4H, ArH), 8.42 (s, 1H, phenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6c) Coumarin), 9.52 (s, 1H, NH), 9.66 (s, 1H, OH) ppm; 13C IR (KBr, cmÀ1): 3400, 1704, 1664, 1624, 1605, 1202, 1063, 844; NMR (125 MHz, DMSO-d6) d: 16.4, 20.3, 26.1, 36.4, 38.0, 1 100.5, 113.4, 116.3, 118.2, 118.6, 124.6, 129.3, 130.7, 134.4, H NMR (500 MHz, DMSO-d6) d: 2.16–2.32 (m, 2H, CH2), 2.58–2.70 (m, 2H, CH ), 4.51 (s, 1H, CH), 7.19 (d, 136.4, 153.2, 154.1, 157.1, 164.5, 195.5 ppm; MS(ESI): m/z 2 + J = 7.2 Hz, 2H, ArH), 7.44 (d, J = 7.2 Hz, 2H, ArH), 7.62– 441 (M+H) ; Anal. Calcd for C25H16N2O6: C, 68.18; H, 7.75 (m, 4H, ArH), 8.33 (s, 1H, Coumarin), 9.39 (s, 1H, 3.63; N, 6.36%. Found: C, 68.11; H, 3.60; N, 6.34%. 13 NH) ppm; C NMR (125 MHz, DMSO-d6) d: 16.6, 20.3, 26.6, 36.2, 37.2, 100.2, 113.1, 116.4, 118.4, 118.9, 124.9, 2.6.7. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(4-hydro 129.3, 130.3, 134.3, 136.4, 153.1, 154.0, 156.3, 163.5, xyphenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6g) 194.5 ppm; MS(ESI): m/z 470 (M+H)+; Anal. Calcd for IR (KBr, cmÀ1): 3458, 3410, 1714, 1664, 1625, 1602, 1216, 1 C25H15N3O7: C, 63.96; H, 3.20; N, 8.95%. Found: C, 63.88; 1073, 840; H NMR (500 MHz, DMSO-d6) d: 2.21–2.28 (m, H, 3.20; N, 2.90%. 2H, CH2), 2.49–2.57 (m, 2H, CH2), 4.52 (s, 1H, CH), 7.20 (d, J = 7.2 Hz, 2H, ArH), 7.38 (d, J = 7.2 Hz, 2H, ArH), 2.6.4. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(4-bromo 7.70–7.82 (m, 4H, ArH), 8.50 (s, 1H, Coumarin), 9.43 (s, 1H, phenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6d) NH), 9.72 (s, 1H, OH) ppm; 13C NMR (125 MHz, DMSO- IR (KBr, cmÀ1): 3415, 1699, 1661, 1618, 1600, 1206, 1070, 840; d6) d: 16.6, 19.9, 26.8, 36.3, 38.2, 100.1, 113.3, 116.5, 118.1, 1 118.9, 124.8, 129.3, 130.8, 134.3, 136.4, 153.1, 154.3, 157.5, H NMR (500 MHz, DMSO-d6) d: 2.04–2.12 (m, 2H, CH2), 164.0, 195.8 ppm; MS(ESI): m/z 441 (M+H)+; Anal. Calcd 2.32–2.46 (m, 2H, CH2), 4.46 (s, 1H, CH), 7.25–7.42 (m, 4H, ArH), 7.78–7.90 (m, 4H, ArH), 8.44 (s, 1H, Coumarin), 9.45 for C25H16N2O6: C, 68.18; H, 3.63; N, 6.36%. Found: C, 13 68.20; H, 3.65; N, 6.33%. (s, 1H, NH) ppm; C NMR (125 MHz, DMSO-d6) d: 16.7, 20.4, 26.0, 36.3, 37.1, 100.4, 113.6, 116.0, 118.4, 119.0, 124.8, 129.3, 131.0, 134.2, 136.4, 152.8, 153.8, 156.8, 163.8, 2.6.8. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(4-N,N-dime 194.9 ppm; MS(ESI): m/z 503.9 (M+H)+; Anal. Calcd for thylaminophenyl)-cyclopenta [b]pyrano-pyrimidine-4,6-5H- C25H15BrN2O5: C, 59.65; H, 2.98; N, 5.56%. Found: C, dione (6h) 59.55; H, 2.95; N, 5.54%. IR (KBr, cmÀ1): 3402, 1712, 1661, 1622, 1601, 1213, 1053, 845; 1 H NMR (500 MHz, DMSO-d6) d: 2.15–2.22 (m, 2H, CH2), 2.6.5. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(4-chloro 2.55–2.62 (m, 2H, CH2), 2.68 (s, 6H, N(CH3)2), 4.48 (s, 1H, phenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6e) CH), 7.16 (d, J = 7.2 Hz, 2H, ArH), 7.42 (d, J = 7.2 Hz, IR (KBr, cmÀ1): 3424, 1708, 1658, 1619, 1607, 1208, 1073, 853; 2H, ArH), 7.65–7.79 (m, 4H, ArH), 8.55 (s, 1H, Coumarin), 1 13 H NMR (500 MHz, DMSO-d6) d: 2.18–2.30 (m, 2H, CH2), 9.40 (s, 1H, NH) ppm; C NMR (125 MHz, DMSO-d6) d: 2.62–2.73 (m, 2H, CH2), 4.39 (s, 1H, CH), 7.10 (d, 15.9, 19.9, 26.6, 36.4, 37.0, 100.9, 113.4, 116.0, 118.1, 118.9, Synthesis of cyclopenta[b]pyrano pyrimidinone derivatives 5

124.7, 129.3, 131.2, 134.3, 136.3, 152.8, 153.9, 156.8, 164.3, Table 1 Preparation of various 2-amino-4-phenyl-5-oxo- 194.6 ppm; MS(ESI): m/z 468 (M+H)+; Anal. Calcd for 4,5,6,7-tetrahydrocyclopenta[b]pyran-3-carbonitrile C H N O : C, 69.38; H, 4.49; N, 8.99%. Found: C, 69.31; 27 21 3 5 derivatives.a H, 4.46; N, 8.97%. Entry R1 Product Time (h) Yield (%)b Mp (C) 2.6.9. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(3-methy 1H 4a 2.0 93 204–206 lphenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6i) 2 4-CH3 4b 2.0 90 206–208 IR (KBr, cmÀ1): 3418, 1703, 1657, 1617, 1607, 1209, 1070, 796; 3 4-NO2 4c 2.0 88 201–203 4 3-Br 4d 1.5 89 220–222 1H NMR (500 MHz, DMSO-d ) d: 2.17–2.29 (m, 2H, CH ), 6 2 5 4-Cl 4e 1.5 90 242–244 2.34 (s, 3H, CH3), 2.52–2.64 (m, 2H, CH2), 4.50 (s, 1H, CH), 6 3-OH 4f 1.5 92 212–214 7.11–7.31 (m, 4H, ArH), 7.73–7.87 (m, 4H, ArH), 8.41 (s, 7 4-OH 4g 2.0 90 228–230 13 1H, Coumarin), 9.38 (s, 1H, NH) ppm; C NMR 8 4-N(CH3)2 4h 2.0 89 236–238 (125 MHz, DMSO-d6) d: 16.2, 20.0, 26.2, 36.1, 37.1, 100.8, 9 3-CH3 4i 1.5 89 198–200 113.7, 115.6, 117.9, 118.7, 124.8, 129.3, 131.4, 135.0, 137.2, 10 3-OCH3 4j 2.0 92 190–192 153.4, 154.1, 156.9, 164.0, 195.5 ppm; MS(ESI): m/z 439 a Reaction conditions: cyclopentane-1,3-dione (1 mmol), aldehy- + (M+H) ; Anal. Calcd for C26H18N2O5: C, 71.23; H, 4.11; de (1 mmol) and malononitrile (1 mmol) in the presence of Alum

N, 6.39%. Found: C, 71.20; H, 4.07; N, 6.33%. (KAl(SO4)2Æ12H2O) (10 mol%) in solvent-free conditions at 70 C. b Isolated yield. 2.6.10. 7,8-Dihydro-2-(2-oxo-2H-chromen-3-yl)-5-(3-metho xyphenyl)-cyclopenta[b]pyrano-pyrimidine-4,6-5H-dione (6j) the amount from 5 to 8 mol% has no effect on the product IR (KBr, cmÀ1): 3406, 1709, 1662, 1626, 1600, 1214, 1029, 847, yield and reaction time (Table 2, entry 8). 769; 1H NMR (500 MHz, DMSO-d ) d: 2.18–2.29 (m, 2H, 6 To expand the generality of this novel catalytic method, CH ), 2.40–2.56 (m, 2H, CH ), 3.66 (s, 3H, OCH ), 4.47 (s, 2 2 3 various cyclopenta[b]pyrano pyrimidinone derivatives 6a–j 1H, CH), 7.18–7.33 (m, 4H, ArH), 7.76–7.90 (m, 4H, ArH), (Scheme 2) were synthesized under the optimized conditions 8.39 (s, 1H, Coumarin), 9.41 (s, 1H, NH) ppm; 13C NMR and the results are presented in Table 3. After completion of (125 MHz, DMSO-d ) d: 16.3, 20.2, 26.3, 35.7, 37.0, 101.2, 6 the reaction the catalyst, SBPDSA was recovered by 114.2, 116.0, 118.3, 118.8, 125.0, 129.3, 131.4, 134.2, 136.4, evaporating the aqueous layer, washed with acetone, dried 152.8, 153.8, 157.4, 166.0, 196.0 ppm; MS(ESI): m/z 455 and reused for subsequent reactions without significant loss (M+H)+; Anal. Calcd for C H N O : C, 68.72; H, 3.96; 26 18 2 6 in its activity (Fig. 1). All the structures of the synthesized N, 6.16%. Found: C, 68.61; H, 3.93; N, 6.15%. compounds 6 were confirmed by their analytical and spectroscopic data. 3. Results and discussion A probable mechanism for the formation of 6a as a model via the condensation reaction is outlined in Scheme 3. Firstly, The synthetic pathway of compounds (6a–j) was achieved via the protonation of coumarin-3-carboxylic acid by SBPDSA as the intermediates 2-amino-4-phenyl-5-oxo-4,5,6,7-tetrahydro- a solid acid occurred to form a cation intermediate (a). In con- cyclopenta[b]pyran-3-carbonitrile derivatives (4a–j). These tinuation, the formation of (b) resulting from the amidation of compounds (4a–j) were obtained by the three component (a) with 4a was established. In the next step, the protonation of condensation of aldehydes 1, malononitrile 2 and cyclopen- nitrile group of intermediate (b) following by a cyclo-addition tane-1,3-dione 3 using Alum (KAl(SO4)2Æ12H2O) (10 mol%), reaction occurred to form the intermediate (c). In continuation À under solvent-free conditions (Scheme 1). Due to its mild the addition reaction of -SO3 followed by ring opening of the and reusable catalytic activity for the synthesis of pyran (c) to the intermediate (d) and (e) followed by ring closure of derivatives (Rajguru et al., 2013), we opted to use Alum intermediate (e) results in the formation of intermediate (f) [KAl(SO4)2Æ12H2O] as a non-toxic catalyst. The aromatic alde- that converts to the (6a) as product by the de-protonation hydes 1 bearing electron-withdrawing and electron donating reaction. Interestingly, the formation of compound 6a, groups were found to be equally effective to produce 2- obtained from the condensation of coumarin-3-carboxylic acid amino-4H-pyrans 4a–j in very good yields (Table 1). with 4a, confirms the mechanism of the reaction which was After the synthesis of 2-amino-4-phenyl-5-oxo-4,5,6,7-te- rarely described in the literature as Dimroth rearrangement trahydrocyclopenta[b]pyran-3-carbonitrile derivatives 4,we (Foucourt et al., 2010; Dimroth, 1909). have synthesized compounds 6. To optimize the reaction The possibility of recycling the catalyst was examined using conditions, the effect of catalyst loading was investigated the condensation reaction of compound 4a with coumarin-3- between compound 4a and coumarin-3-carboxylic acid. The carboxylic acid 5 in the optimized conditions. The catalyst reaction was carried out under neat conditions at 80 C with- was recovered from the aqueous phase (filtration), washed out and with different acid catalysts (cellulose sulfuric acid, with acetone, dried and re-used for subsequent reactions with- silica sulfuric acid, sulfamic acid, SBPDSA each 5 mol%). out loss of activity and efficiency. The recycled catalyst could The maximum yield was obtained using SBPDSA. It can be be reused five times without any additional treatment or appre- seen that the reaction did not proceed even after 12 h in the ciable reduction in catalytic activity (Fig. 1). absence of this catalyst (Table 2, entry 1). Although a lower X-ray diffraction (XRD) for SBPDSA using powder X-ray catalyst loading of 3 or 2 mol% accomplished this condensa- diffraction measurements was performed using Advance tion, 5 mol% of SBPDSA was optimal in terms of reaction diffractometer made by Bruker AXS company in Germany. time and isolated yield (Table 2, entries 5 and 6). Increasing Scans were taken with a 2h step size of 0.04 and a counting 6 P.N. Sudhan, S. Sheik Mansoor

Table 2 Preparation of 7,8-dihydro-2-(2-oxo-2H-chromen-3-yl)-5-phenyl-cyclopenta[b] pyrano-pyrimidine-4,6-5H-dione: Effect of catalyst.a Entry Catalyst Amount of catalyst (mol%) Time (h) Yield (%)b 1 None 0 12.0 Trace 2 Cellulose sulfuric acid 5 6.0 70 3 Silica sulfuric acid 5 6.0 73 4 Sulfamic acid 5 8.0 59 5 SBPDSA 5 (0.051 g) 4.0 90 6 SBPDSA 8 (0.081 g) 4.0 90 7 SBPDSA 3 (0.031 g) 4.0 77 8 SBPDSA 2 (0.021 g) 4.0 69 a Reaction conditions: 4a (1 mmol) and coumarin-3-carboxylic acid (1 mmol) at 80 C. b Isolated yield.

Table 3 Preparation of various 7,8-dihydro-2-(2-oxo-2H-chromen-3-yl)-5-aryl-cyclopenta [b]pyrano-pyrimidine-4,6-5H-dione derivatives.a Entry Compound 4 Product Time (h) Yield (%)b Mp (C) 1 4a 6a 4.0 90 266–268 2 4b 6b 4.0 88 224–226 3 4c 6c 3.5 91 258–260 4 4d 6d 3.5 91 234–236 5 4e 6e 3.5 90 238–240 6 4f 6f 4.0 90 246–248 7 4g 6g 4.0 92 254–256 8 4h 6h 4.0 88 262–264 9 4i 6i 4.0 89 244–246 10 4j 6j 4.0 89 272–274 a Reaction conditions: 4a–j (1 mmol), and coumarin-3-carboxylic acid (1 mmol) in the presence of SBPDSA (5 mol%) at 80 C. b Isolated yield.

94 in Fig. 2 no significant change in the structure of catalyst Recycleability of SBPDSA was observed during the reaction. 92 90% 3.1. Biological evaluations 90 88% Variously substituted on the aromatic ring, the compounds 88 87% 6a–j may be useful in understanding the influence of steric 86 and electronic effects on biological activity. They were tested Yield (%) 85% for their antibacterial and antifungal activity at different con- 84% 84 centrations in DMSO. Ciprofloxacin and Amphotericin-B were used as the positive control drugs for antibacterial and 82 antifungal tests, respectively. Inoculums of the bacterial and fungal cultures were also prepared. The minimum concentra- 80 tion at which no growth was observed was taken as the 12345 minimum inhibitory concentration (MIC) value. Number of Runs 3.2. Antibacterial activity Figure 1 Recycling of catalyst SBPDSA for the synthesis of 7,8- dihydro-2-(2-oxo-2H-chromen-3-yl)-5-phenyl-cyclopenta[b]pyra- no-pyrimidine-4,6-5H-dione from 4a and coumarin-3-carboxylic The newly synthesized compounds were screened for their in vit- acid. ro antibacterial activity against Escherichia coli, Pseudomonas aeruginosa and Klebsiella pneumoniae bacterial strains by the serial plate dilution method. Serial dilutions of the drug in Mul- time of 30 s at room temperature. Specimens for XRD were ler Hinton broth were taken in tubes and their pH was adjusted prepared by compaction into a glass-backed aluminum sample to 5.0 using phosphate buffer. A standardized suspension of the holder. Data were collected over a 2h range from 4 to 75. The test bacterium was inoculated and incubated for 16–18 h at fresh catalyst and recovered catalyst were characterized by 37 C. The MIC is the lowest concentration of the drug for XRD and their pattern is presented in Fig. 2. As it is shown which no growth is detected. The results are summarized in Synthesis of cyclopenta[b]pyrano pyrimidinone derivatives 7

O H O SO 3H SO 3 O HO CN HO + O O O O O NH2 a 4a SBPDSA = SO3H

Heat -H O 2 SO3H O O NH SO3 SO3H CN O O Heat O N O N b H c H O O O O

SO3

O NH O O O Heat NH O SO2 O SO2 O N H O N d e H O O O O

SO3

O O O O SO3H SO3 NH NH O N O N H f 6a O O O O

Scheme 3 A possible mechanism for the formation of 7,8-dihydro-2-(2-oxo-2H-chromen-3-yl)-5-aryl-cyclopenta[b] pyrano-pyrimidine- 4,6-5H-dione derivatives.

Figure 2 XRD pattern of fresh and recovered SBPDSA.

Table 4. The MIC values were evaluated at concentration range, found to have antibacterial activity against E. coli, P. aerugi- 12.5–25 lg/mL. Upon exploration of the antibacterial activity nosa and K. pneumoniae, and Staphylococcus aureus when com- data (Table 4), it has been observed that all compounds were pared with the employed standard drug. 8 P.N. Sudhan, S. Sheik Mansoor

Table 4 In vitro antibacterial and antifungal activities of compounds 6a–j. Compounds Minimum inhibitory concentration (MIC) in lg/mL Antibacterial activity Antifungal activity E. coli P. aeruginosa K. pneumonia A. flavus R. schipperae A. niger 6a 150 100 150 150 150 150 6b 100 50 75 125 75 75 6c 25 25 25 50 25 50 6d 50 50 50 50 50 50 6e 75 50 75 100 75 100 6f 75 75 75 100 100 100 6g 150 150 150 150 150 150 6h 150 100 150 150 150 150 6i 75 75 75 100 75 100 6j 75 50 75 100 75 100 Ciprofloxacin 25 12.5 25 – – – Amphotericin-B – – – 50 25 50

3.3. Antifungal activity 3.5. Acute toxicity

Newly prepared compounds were also screened for their anti- The median lethal doses (LD50) of the synthesized compounds fungal activity against Aspergillus flavus, Rhizopus schipperae 6a–j were determined in mice (Sztaricskai et al., 1999). Groups and Aspergillus niger in DMSO by the serial plate dilution of male adult mice, each of six animals, were injected i.p. with method. Sabourauds agar media were prepared by dissolving graded doses of each of the test compounds. The percentage of peptone (1 g), D-glucose (4 g) and agar (2 g) in distilled water mortality in each group of animals was determined 24 h, after (100 mL) and adjusting the pH to 5.7. Normal saline was used injection. Computation of LD50 was processed by a graphical to make a suspension of sore of fungal strains for lawning. method. The LD50 values for 6a–j were 20–30 times higher Activity of each compound was compared with Ampho- than its MIC. tericin-B as standard. The results are summarized in Table 4. The MIC values were evaluated at concentration range, 4. Conclusions 25–50 lg/mL. The results given in Table 4 show that all com- pounds exhibited antifungal activity with MIC against A. flavus, We have developed a green and simple protocol for the synthe- R. schipperae and A. niger compared with Amphotericin-B as sis of 7,8-dihydro-2-(2-oxo-2H-chromen-3-yl)-5-phenyl-cy- standard drug. clopenta[b]pyrano-pyrimidine-4,6-5H-dione derivatives via the condensation of 2-amino-4-phenyl-5-oxo-4,5,6,7-tetrahy- 3.4. Influence of aromatic substituents drocyclopenta[b]pyran-3-carbonitrile derivatives and coumar- in-3-carboxylic acid using SBPDSA as an efficient novel The results suggest that the antibacterial and antifungal activ- catalyst. This procedure is a promising strategy and has advan- ities are markedly influenced by the aromatic substituents. tages such as easy workup and eco-friendliness. It is expected Compound 6a without any substituent in the aryl moiety exhi- that the present methodology will find application in organic bits antibacterial activity in vitro at 150, 100 and 150 lg/ml synthesis. All the synthesized compounds were screened for against E. coli, P. aeruginosa and K. pneumonia respectively their in vitro antimicrobial activity compared to the standard and also exhibits antifungal activity in vitro at 150, 150 and drug Ciprofloxacin and Amphotericin-B for antibacterial and 150 lg/ml against A. flavus, R. schipperae and A. niger, respec- antifungal activity respectively. tively. Compounds 6c, 6d and 6e with electron-withdrawing substituents in the aromatic ring show greater antibacterial Acknowledgments activity than the other compounds against all the tested organ- isms. Also, compounds 6c and 6d show greater antifungal activ- The authors are thankful to the Management of C. Abdul ity than all the other compounds against all the tested Hakeem College, Melvisharam – 632 509, Tamil Nadu, India organisms. The aromatic substituents in 6c and 6e have positive for the facilities and support. values for the Hammett substituent constant rp [NO2 (+0.78) and Cl (+0.23)] and the aromatic substituent in 6d also has References positive value for the Hammett substituent constant rm [Br (+0.40)]. The aromatic substituents in 6b and 6h have negative Asadipour, A., Alipour, M., Jafari, M., Khoobi, M., Emami, S., values for the Hammett substituent constant r [CH (À0.17) p 3 Nadri, H., Sakhteman, A., Moradi, A., Sheibani, V., Moghadam, and N(CH ) ( 0.205)] and the aromatic substituent in 6i also 3 2 À F.H., Shafiee, A., Foroumadi, A., 2013. Novel coumarin-3- has negative value for the Hammett substituent constant rm carboxamides bearing N-benzylpiperidine moiety as potent acetyl- [CH3 (À0.069)]. The Hammett substituent constant r for the cholinesterase inhibitors. Eur. J. Med. Chem. 70, 623–630. aromatic substituents in 6f and 6g is rm OH (+0.12) and rp Augustine, J.K., Bombrun, A., Ramappa, B., Boodappa, C., 2012. An OH (À0.37), respectively. Hence, 6f is more active than 6g. efficient one-pot synthesis of coumarins mediated by propylphos- Synthesis of cyclopenta[b]pyrano pyrimidinone derivatives 9

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University of Bahrain Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com

ORIGINAL ARTICLE Synthesis, characterization and in vitro drug release of cisplatin loaded Cassava starch acetate–PEG/ gelatin nanocomposites

V. Raj *, G. Prabha

Advanced Materials Research Laboratory, Department of Chemistry, Periyar University, Salem 11, Tamil Nadu, India

Received 27 April 2015; revised 6 August 2015; accepted 23 August 2015 Available online 19 November 2015

KEYWORDS Abstract The aim of the present study is to examine the feasibility of Cassava starch acetate Cassava starch acetate; (CSA)–polyethylene glycol (PEG)–gelatin (G) nanocomposites as controlled drug delivery systems. Drug delivery; It is one of the novel drug vehicles which can be used for the controlled release of an anticancer Polyethylene glycol (PEG); drug. Simple nano precipitation method was used to prepare the carriers CSA–PEG–G nanocom- Gelatin; posites and they were used for entrapping cisplatin (CDDP). Through FT-IR spectroscopy, the Cisplatin; linking among various components of the system was proved and with the help of scanning electron Nanocomposites microscope and transmission electron microscopy (TEM), the surface morphology was investigated. The particle sizes of the CSA–CDDP, CSA–CDDP–PEG and CSA–CDDP–PEG–G polymer com- posites were between 140 and 350 nm, as determined by a Zetasizer. Drug encapsulation efficiency, drug loading capacity and in vitro release of CDDP were evaluated respectively. The findings revealed that the cross linked CSA–PEG–G nanocomposites can be a potential polymeric carrier for controlled delivery of CDDP. Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction One of the efficacious methods, applied in this study to improve the properties of starch, is the chemical modification Starch, a biodegradable polymer is a promising carrier for of starch which includes esterification. Over the past two dec- drug delivery. It has been used in various fields like biomedical, ades extensive studies have been conducted on starch ester agriculture and food etc. However, native starch cannot fit into called as acetylated starch (Wang and Wang, 2002). Chemi- some parental controlled drug delivery systems, as many drugs cally transformed starch acetates are less hydrophilic than are released quickly from such unmodified starch-based sys- most of the other modified starches, due to the hydrophobic tems (Michailova et al., 2001); due to considerable swelling nature of the acetoxy substituent (OCOCH3). In the drug and quick enzymatic degradation of native starch in biological delivery applications, starch acetate has been extensively used systems. (Korhonen et al., 2004; Nutan et al., 2005, 2007; Pajander et al., 2008; Pohja et al., 2004; Pu et al., 2011; Tuovinen * Corresponding author. Tel.: +91 9790694972, +91 9789703632. et al., 2004a,b; van Veen et al., 2005; Xu et al., 2009) and tissue E-mail address: [email protected] (V. Raj). engineered scaffold has also been investigated (Guan and Peer review under responsibility of University of Bahrain. Hanna, 2004; Reddy and Yang, 2009). http://dx.doi.org/10.1016/j.jaubas.2015.08.001 1815-3852 Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Drug release of cisplatin loaded cassava starch acetate 11

In cancer chemotherapy, platinum compounds play an The reaction was carried out at 120 °C for a period of 3 h. important role. One of the most common anticancer agents The final product was precipitated with ethanol, filtered and is Cisplatin (CDDP), the first generation of platinum based dried in vacuum oven. Lastly, the modified starch was milled chemotherapy drug. It is used in the treatment of solid tumors and sifted in a sieve (#50 mesh) to obtain a homogeneous par- including gastrointestinal, head and neck, genitourinary and ticle size and stored in desiccators until further study. lung tumors (Kelland, 2007; Boulikas and Vougiouka, 2003). The clinical application of cisplatin for cancer chemother- 2.3. Preparation of CSA–CDDP nanorods apy is still in limited use because of its nonspecific bio- distribution and severe side effects. In an attempt to overcome The CSA nanorods were prepared by a simple nanoprecipita- this shortcoming, various studies have been conducted by tion technique as reported by Chin et al., 2011 with slight mod- many groups. They are magnetically mediated controlled ification. CSA (10 mg) was dissolved in 8:10 wt% of NaOH/ delivery systems (Likhitkar and Bajpai, 2012), click chemistry urea (NU) solution mixtures; this solution mixture was used (Huynh et al., 2011), SiO2/polymer for the controlled release of as a solvent system for the dissolution of acetylated cassava cisplatin (Czarnobaj and Lukasiak, 2007), platinum-tethered starch. Cisplatin was dissolved in CSA solution and prepared gold nanoparticle (Brown et al., 2010). Chemotherapy with cis- at various concentrations i.e., 10%, 20%, 30%, 40% and platin is connected with some serious side effects, such as: 50%, using 4, 8, 12, 16 and 20 mg of drug, respectively. An ali- vomiting, nephrotoxicity, ototoxicity, neuropathy, anemia quot of CSA solution (10 mg/mL) containing the various con- and nausea (Uchino et al., 2005). Owing to these side effects centrations of the drugs was added drop-wise into a 10 ml of other methods of administering cisplatin are required. In the absolute ethanol solution, which was constantly stirred using present study, CSA/PEG/G has been chosen as the raw mate- a magnetic stirrer at a constant stirring rate (1500 rpm). The rial to prepare the drug carrier. CSA nanorods were made immediately. This dispersion of The key objective of the current study is to encapsulate the anti- nanorods was vacuum evaporated to remove the organic sol- cancer drug cisplatin (CDDP) into Cassava starch acetate/poly- vent fully. Finally the resultant mixture was centrifuged at ethylene glycol/gelatin (CSA/PEG/G) nanocomposites through 13,000 rpm and the supernatant was removed to obtain the the interaction between cisplatin (CDDP) and CSA/PEG/G CSA-cisplatin nanorods and freeze-dried at 40 °C for 20 h. nanocomposites. Gelatin is a naturally occurring biodegradable macromolecule with well-documented biocompatible properties 2.4. Preparation of the CSA–CDDP–PEG and CSA–CDDP– over other synthetic polymers that make it an appropriate material PEG–G nanocomposites to be used as a nanoparticulate carrier (Lai et al., 2006). To develop the microspheres, nanoparticles and polymers, Polyethy- The various percentage of encapsulated CSA–CDDP in the lene glycol (PEG), a suitable graft-forming polymer, has been PEG and G solution were prepared by a method described extensively employed in pharmaceutical and biomedical fields in our previous report (Rajan et al., 2013) as follows. First, (Jeong et al., 2008). The viability of CDDP-loaded polymeric 10% of PEG solution was prepared in water. Then, the solu- nanocomposites as a drug delivery system was verified by tion was gradually added to a correct portion of the CSA– evaluating its in vitro studies and instrumental characteristics. CDDP nanorods under constant magnetic stirring at room temperature for 1 h. The resulting encapsulated nanocompos- 2. Materials and methods ites (CSA–CDDP–PEG) were collected by centrifugation at 1500 rpm and freeze-dried at 30 °C for 20 h. Later gelatin 2.1. Materials (20 mg) was dissolved in water in a similar manner and grad- ually added to CSA–CDDP–PEG nanocomposites under con- Native Cassava starch powder was obtained from Sago Serve stant magnetic stirring at room temperature for 1 h. Finally the Industries (Salem, India). Acetic acid (P99%) and acetic resulting encapsulated nanocomposites (CSA–CDDP–PEG– anhydride (P98%) were of analytical grade procured from G) were collected by centrifugation at 1500 rpm and freeze- Sigma–Aldrich (St. Louis, USA). PEG 10000, gelatin Type- dried at 30 °C for 20 h. B, phosphate-buffered saline (PBS) were prepared in deionized 3 water using NaCl (0.14 M), KCl (2.68 10 M), Na2HPO4 2.5. Particle size analysis 3 (0.01 M), KH2PO4 (1.76 10 M). Sodium hydroxide (NaOH) and absolute ethanol were purchased from Merck Drug loaded polymeric nanocomposites were characterized for (Mumbai, India Ltd). Cisplatin was obtained from Dabur the particle size, size distribution and zeta potential using Zeta- Pharma Ltd. (New Delhi, India). All chemicals were used with- sizer (Malvern Instruments, UK). out additional purification. 2.6. Scanning electron microscopy (SEM) and Fourier 2.2. Preparation of cassava starch acetate transform infrared spectroscopy (FT-IR) analysis

Native cassava starch was permitted to react with acetic anhy- Morphological characteristics of the freshly prepared (CSA– dride (1:4 ratio) with pyridine as a catalyst as previously CDDP, CSA–CDDP–PEG, and CSA–CDDP–PEG–G) described (Singh and Nath, 2012) with few modifications. nanocomposites were viewed using scanning electron micro- Before acetylation, cassava starch was dried in an oven for scopy (SEM-Hitachi-S-2700), FT-IR spectrum was taken to 20 h at 45–60 °C. Dried starch (25 g) was mixed with acetic study the interaction between polymers and drug using Perkin anhydride (100 g) through the medium of pyridine (200 g). Elmer spectrum RXI. KBr pellets were concisely prepared by 12 V. Raj, G. Prabha mixing 1 mg of the sample with 200 mg of KBr. Fourier Trans- nanocomposites increases slightly with an increase in the % form Infrared spectroscopy (400–4000 cm1) was performed of CDDP encapsulation. The size of the nanocomposites, with a resolution of 2 cm1. which is increased once again, is due to the coordination of PEG and G with CSA–CDDP. The 10% of CDDP loaded 2.7. TEM analysis CSA, CSA–PEG, CSA–PEG–G nanocomposites displayed a mean particle size value of 143,239 and 311 nm respectively The shape and morphology of the Cisplatin loaded CSA, as shown in (Table 1). CSA–PEG and CSA–PEG–G nanocomposites were investi- Zeta potential tells about the charge on the surface of the gated by transmission electron microscopy (TEM, Hitachi polymeric nanocomposites and plays an important role in the H-600-II) operated at 200 kV. stability of the particles in suspension through the electrostatic repulsion between the particles (Wilson et al., 2011). The repul- 2.8. Determination of encapsulation efficiency (EE) and loading sion among the polymeric nanocomposites with the same type of capacity (LC) surface charge provides extra stability (Zhao et al., 2010). The 10% of CDDP loaded CSA, CSA–PEG, CSA–PEG–G nanocomposites exhibited a mean zeta potential value of The suspensions of the drug-loaded polymeric nanocomposites 24.6 mV, 15.3 mV and 10.0 mV, respectively (Table 1) that were centrifuged at 17,000 rpm for 40 min and the EE and LC lies in the stable range indicating that the prepared nanocom- of drug loaded polymeric nanocomposites were determined by posite systems were stable. The negative values obtained for quantifying the absorption of the clear supernatant using a the zeta potential indicate that the polymeric nanocomposites UV-spectrophotometer (Elico SL 159, India). The correspond- surface is negatively charged. This negative charge may be due ing calibration curves were made by testing the supernatant of to the availability of the free acetyl groups on the polymer. blank polymeric nanocomposites. Tests were performed in Negative zeta potential values are detected in all cases, suggest- triplicate for each sample. The absorbance value of CDDP ing that CSA chains are primarily located on the surface of the was measured using a UV–vis spectrophotometer at the wave- particles. All the zeta potential experiments were done in the length of 290 nm. The percentage of encapsulation efficiency aqueous medium after centrifuging the nanocomposites at and loading capacity of CDDP in the CSA, CSA–PEG and 15,000 rpm for 30 min and dispersed in millipore water. CSA–PEG–G nanocomposites are determined by the follow- ing equations (Eqs. (1) and (2)), respectively, as reported ear- 3.1.2. Fourier transmission infrared spectroscopy (FT-IR) lier (Papadimitriou et al., 2008), which are as follows: analysis = % EE ¼ðWt WfÞ Wt 100 ð1Þ The FT-IR spectra of native and acetylated cassava starch are given in Fig. 1. In the spectrum of native starch, there are some = % LC ¼ðWt WfÞ Wn 100 ð2Þ discernible absorbencies at 1157, 1016 cm1, which are attribu- where Wt is the total amount of CDDP; Wf is the amount of ted to C–O bond stretching (Goheen and Wool, 1991). Other free CDDP in the supernatant after centrifugation; and Wn characteristic absorption bands at 928, 859, 765, and is the weight of polymeric nanocomposites after freeze- 576 cm1 are due to the whole anhydroglucose ring stretching drying. All measurements were made in triplicate and the aver- vibrations (Cherif Ibrahima Khalil et al., 2011). The very broad age value was reported. band between 3000–3600 cm1 and 2931 cm1 corresponds to OH and CH stretching respectively (Kacurakova and Wilson, 2.9. Evaluation of in vitro drug release 2001) while the peaks at 1648 cm1 and 1420 cm1 correspond to d (OH) and d (CH) bendings (Mano et al., 2003). Compared to native starch, starch acetates had a strong absorption band The in vitro drug release tests were carried out on all formula- 1 tions (2%, 4%, 6%, 8%, and 10% drug loaded samples). at 1730 cm that is attributed to the stretching vibration of ‚ Nearly 0.1 mg of each sample was suspended in a definite vol- the ester carbonyl C O and indicated the acetylation of starch. ume (10 ml) of phosphate buffer saline (PBS) at various pH at FT-IR spectra of various CSA–CDDP, CS–CDDP–PEG, 37 °C. The resulting suspension was placed in an incubated and CSA–CDDP–PEG–G nanopolymer composites are shown in Fig. 2a. The CSA spectra exhibited band at shaker at 120 rpm for a definite time period (1 h) and five- 1 milliliter aliquots were taken out of the dissolution medium 1730 cm relative to the stretching of the ester carbonyl (C‚O) group of acetylated cassava starch. The addition of at appropriate time intervals (30 min), replaced by same vol- 1 ume of fresh PBS buffer, to keep the volume of the release PEG and gelatin led to bands at 1666 cm relative to the medium constant. The amount of drug released was observed stretching of the ester carbonyl group (Lin et al., 2007). Thus by UV spectrophotometer (Systronics, India) at 290 nm. the shift in the peak toward a lower field than the actual field indicates the physical mixture of CSA, PEG, and gelatin. The amino peak of gelatin in the polymer composite was shifted 3. Results and discussion from 1542 to 1547 cm–1. The amide I absorption was primarily due to the stretching vibration of the C–O bond and the amide 3.1. Characterization of polymeric nanocomposites II band was due to the coupling of the bending of the N–H bond and the stretching of the C–N bond. This result proves 3.1.1. Preparation and characterization of CDDP loaded that there is an interaction between the CSA and the amino polymeric nanocomposites groups of gelatin. Table 1 represents the particle size and zeta potentials of the Furthermore, strong characteristic peaks of CDDP are not CDDP loaded polymeric nanocomposites. The size of the sensed at the same position in the drug-loaded nanocompos- Drug release of cisplatin loaded cassava starch acetate 13

Table 1 The particle size and zeta potential values of CSA–CDDP, CSA–CDDP–PEG and CSA–CDDP–PEG–G. % of CDDP concentration Particle size (nm) mean ± SDa ZP (mV) mean SDa CSA–CDDP CSA–CDDP–PEG CSA–CDDP–PEG–G CSA–CDDP CSA–CDDP–PEG CSA–CDDP–PEG–G 10 143.2 ± 12.5 239.0 ± 11.8 311.7 ± 05.0 24.6 ± 1.1 15.3 ± 1.4 10.0 ± 2.0 20 154.4 ± 12.7 245.5 ± 05.8 325.4 ± 10.1 22.3 ± 1.5 13.5 ± 1.7 08.1 ± 1.1 30 168.0 ± 08.6 261.3 ± 15.3 333.6 ± 12.4 20.1 ± 2.2 13.0 ± 1.5 07.3 ± 3.2 40 185.5 ± 11.3 270.1 ± 10.1 345.3 ± 10.3 18.8 ± 1.5 11.2 ± 2.3 06.1 ± 1.4 50 193.2 ± 12.5 274.8 ± 13.6 350.1 ± 07.2 19.5 ± 1.8 10.4 ± 1.4 05.5 ± 1.5 CDDP: Cisplatin; CSA: Cassava starch acetate; PEG: polyethylene glycol; G: gelatin; SD: standard deviation for three determinations. a n = 3. The experiments were repeated twice.

1.0 (a) a 100 0.8

80 0.6

0.4 60

0.2 40

0.0 Transmittance (%) 20 CSA-CDDP 1.0 b CSA-CDDP-PEG 0 CSA-CDDP-PEG-G 0.8 1000 2000 3000 4000 Transmittance (%) 0.6 Wavenumber cm-1

0.4 (b) 100

0.2 80 0.0 0 1000 2000 3000 4000 5000 60 Wavenumber cm-1

Figure 1 FTIR spectra of (a) acetylation of cassava starch, (b) 40 native cassava starch. Transmittance (%)

20 ites, identifying an interaction between the drug and the poly- 10% CSA-CDDP-PEG-G mer composites. Also, the peaks were shifted toward a lower 50% CSA-CDDP-PEG-G field than the actual field, and this shift was due to the hydro- 0 gen bond in the encapsulated polymer composites. Fig. 2b 1000 2000 3000 4000 indicates the FT-IR spectra of the 10 and 50% of CDDP Wavenumber cm-1 coated CSA–CDDP–PEG–G polymeric nanoparticles. The spectra of the coated polymeric nanoparticles displayed the Figure 2 FTIR spectra of (a) and (b) cisplatin loaded same peaks that vary only in intensity. nanocomposites.

3.1.3. Scanning electron microscopy (SEM) of drug loaded polymeric nanocomposites can be seen in the SEM of the secondary mixed-film due to The SEM images of CSA–CDDP, CSA–CDDP–PEG and C the physical mixture of PEG and G to the CSA. The mixture SA–CDDP–PEG–G nanocomposites are shown in Fig. 3. of CSA with CDDP exhibited certain immiscibility. A clear In addition, the combination of CDDP with nanocompos- uniform surface of the G cross-linked polymer composite is ites produced a smooth surface and compact structure shown in Fig. 3c. The coating lying of CDDP with CSA Fig. 3b and c. Particle accumulation and a smooth surface exposed a rough surface. The coating lying of PEG with gela- 14 V. Raj, G. Prabha

Figure 3 SEM images of cisplatin loaded (a) CSA, (b) CSA–PEG, (c) CSA–PEG–G nanocomposites.

Figure 4 TEM images of cisplatin loaded (a) CSA, (b) CSA–PEG, (c) CSA–PEG–G nanocomposites.

Table 2 Encapsulation efficiency (EE) and loading capacity (LC) of CSA–CDDP, CSA–CDDP–PEG and CSA–CDDP–PEG–G nanocomposites. % of CDDP concentration CSA–CDDP nanocomposites CSA–CDDP–PEG nanocomposites CSA–CDDP–PEG–G nanocomposites % of EE % of LC % of EE % of LC % of EE % of LC 10 23.9 05.3 45.3 10.1 66.9 14.9 20 62.3 22.7 75.5 27.4 84.0 30.5 30 75.4 34.8 85.5 39.4 90.4 41.7 40 82.0 43.8 89.4 47.7 93.6 49.2 50 85.9 50.6 91.6 53.9 94.1 57.0 tin revealed a smoother surface and rod shaped structure, dis- initial concentration of CDDP is same, which might be accred- playing an enhanced encapsulation efficiency and a more con- ited to the fact that CSA–PEG–G nanocomposites have more sistent structure. attraction than CSA and CSA–PEG nanocomposites with CDDP. The drug encapsulation efficiency (EE) and drug load- 3.1.4. Transmission electron microscopy (TEM) analysis ing capacity (LC) of high percentage of (50%) CDDP loaded The surface morphology of the Cisplatin loaded CSA, CS– CSA, CSA–PEG & CSA–PEG–G nanocomposites were found PEG, and CSA–PEG–G nanocomposites was observed by to be 85.9%, 91.6% & 94.1% and 50.6%, 53.9%, &57.0%, TEM as shown in Fig. 4. Fig. 4a–c illustrates that the CSA– respectively (Table 2). CDDP, CSA–CDDP–PEG, CSA–CDDP–PEG–G nanocom- posites have spherical morphology and are homogeneously 3.3. In vitro drug release studies distributed with an average diameter of 50–100 nm. In vitro drug release studies were done via direct dispersion 3.2. Encapsulation efficiency (EE) and loading capacity (LC) method as explained in the literature (Bisht et al., 2007; Anitha et al., 2011) at pH 3.4 & 7.4 and release pattern is The initial concentration of CDDP played a significant role in shown in Fig. 5. The percentage release of CDDP from CSA deciding the drug encapsulation efficiency (EE) and drug load- was slightly greater when compared to that of CSA–PEG ing capacity (LC) of the CSA, CSA–PEG and CSA–PEG–G and CSA–PEG–G combined nanocomposites. For the CSA– nanocomposites as shown in Table 2. When the concentration CDDP, CSA–CDDP–PEG and CSA–CDDP–PEG–G coated of CDDP is increased, the EE & LC of CSA, CSA–PEG and nanocomposites, the percentage of CDDP released from the CSA–PEG–G nanocomposites is also increased. The EE & nanocomposites was initially much larger and then very slow LC of CSA–PEG–G nanocomposites is somewhat more than after some hours, similar to that reported by other authors that of the CSA and CSA–PEG nanocomposites when at the (Li et al., 2008; Chen et al., 2009, 2011; Yang et al., 2008; Drug release of cisplatin loaded cassava starch acetate 15

Figure 5 In vitro analysis of cisplatin encapsulated nanocomposites.

Zhang et al., 2009). Fig. 5 shows the release profile at both the sary instrumental facilities and the dedicated support of C. pH for the same drug loading, as the pH value of the releasing Vijayabhaskar. buffer increased, the releasing rate of CDDP increased. From Fig. 5, we also found that the release rate of the CDDP which References was loaded in the CSA, CSA–PEG and CSA–PEG–G loaded nanocomposites was much lower than the free CDDP. The Anitha, A., Maya, S., Deepa, N., Chennazhi, K.P., Nair, S.V., results indicated that the release of CDDP from CSA, CSA– Tamura, H., 2011. Efficient water-soluble biodegradable polymeric PEG and CSA–PEG–G nanocomposites is pH dependant, nanocarrier for the delivery of curcumin to cancer cells. Carbohydr. the CDDP released faster in acidic environment than at basic Polym. 83, 452–461. environment as a consequence of binding between drug and Bisht, S., Feldmann, G., Soni, S., Ravi, R., Karikar, C., Maitra, M., the carboxyl group in cassava starch acetate nanocomposites 2007. Polymeric nanoparticle-encapsulated curcumin (‘‘nanocur- cumin”): a novel strategy for human cancer therapy. J. which could be recovered by the attacking of H+ or Cl.At Nanobiotechnol. 5, 3. http://dx.doi.org/10.1186/1477-3155r-r5-3. body environment, the Cl concentration is very high (95– Boulikas, T., Vougiouka, M., 2003. Cisplatin and platinum drugs at 105 mM) and relatively stable in body circulation, more acidic the molecular level. Oncol. Rep. 10, 1663–1683. + environment means more H which can speed up the release Brown, S.D., Nativo, P., Smith, J.A., Stirling, D., Edwards, P.R., of CDDP from the coated polymeric nanocomposites. Venugopal, B., Flint, D.J., Plumb, J.A., Graham, D., Wheate, N.J., 2010. Gold nanoparticles for the improved anticancer drug delivery 4. Conclusions of the active component of oxaliplatin. J. Am. Chem. Soc. 132 (13), 4678. Chen, H.L., Yang, W.Z., Chen, H., Liu, L.R., Gao, F.P., Yang, X.D., In this study, a novel formulation of CDDP loaded CSA, 2009. Surface modification of mitoxantrone-loaded PLGA nano- CSA–PEG, CSA–PEG–G nanocomposites was successfully spheres with chitosan. Colloids Surf. B 73, 212–218. developed and characterized. Size and shape of the prepared Chen, M., Liu, Y., Yang, W., Li, X., Liu, L., Zhou, Z., 2011. nanocomposites were examined using SEM and TEM. Also, Preparation and characterization of self-assembled nanoparticles of the various suspended groups present in the composites have 6-O-cholesterol-modified chitosan for drug delivery. Carbohydr. been determined through the FT-IR studies. The nanocompos- Polym. 84, 1244–1251. ites showed pH and time dependent drug release as confirmed Cherif Ibrahima Khalil, D., Hai Long, L., Bi Jun, X., John, S., 2011. by the in vitro drug dissolution profiles. Drug penetration and Effects of acetic acid/acetic anhydride ratios on the properties of corn starch acetates. Food Chem. 26, 1662–1669. in vitro tests suggest that further study is required to develop Chin, S.F., Pang, S.C., Tay, S.H., 2011. Size controlled synthesis of an in vivo drug delivery system. These results suggest that starch nanoparticles by a simple nanoprecipitation method. Car- the CDDP coated CSA, CSA–PEG and CSA–PEG–G bohydr. Polym. 86, 1817–1819. nanocomposites might be used as great potential carriers for Czarnobaj, K., Lukasiak, J., 2007. In vitro release of cisplatin from controlled drug delivery system. sol–gel processed organically modified silica xerogels. J. Mater. Sci. – Mater. Med. 18 (10), 2041. Acknowledgements Goheen, S.M., Wool, R.P., 1991. Degradation of polyethylene starch blends in soil. J. Appl. Polym. Sci. 42, 2691–2701. One of the authors (G. Prabha) would like to acknowledge the Guan, J., Hanna, M.A., 2004. Extruding foams from corn starch acetate and native corn starch. Biomacromolecules 5 (6), 2329– National Centre for Nanoscience and Nanotechnology, 2339. University of Madras, Chennai, India for providing the neces- 16 V. Raj, G. Prabha

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ORIGINAL ARTICLE Efficient adsorption of 4-Chloroguiacol from aqueous solution using optimal activated carbon: Equilibrium isotherms and kinetics modeling

Afidah Abdul Rahim a, Zaharaddeen N. Garba a,b,* a School of Chemical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia b Department of Chemistry, Ahmadu Bello University, P.M.B. 1044, Zaria, Nigeria

Received 28 July 2015; revised 31 August 2015; accepted 3 September 2015 Available online 29 November 2015

KEYWORDS Abstract The optimal activated carbon produced from Prosopis africana seed hulls (PASH-AC) ° Prosopis africana seed hulls; was obtained using the impregnation ratio of 3.19, activation temperature of 780 C and activation 2 Activated carbon; time of 63 min with surface area of 1095.56 m /g and monolayer adsorption capacity of 498.67 mg/g. Isotherms and kinetics mod- The adsorption data were also modeled using five various forms of the linearized Langmuir eling; equations as well as Freundlich and Temkin adsorption isotherms. In comparing the legitimacy Adsorption; of each isotherm model, chi square (v2) was incorporated with the correlation coefficient (R2)to 4-Chloroguiacol justify the basis for selecting the best adsorption model. Langmuir-2 > Freundlich > Temkin isotherms was the best order that described the equilibrium adsorption data. The results revealed pseudo-second-order to be the most ideal model in describing the kinetics data. Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Adsorption is one of the most useful and effective among the control technologies (El Haddad et al., 2012, 2013; Recently, one of the most troubling environmental challenges Noreen and Bhatti, 2014; Noreen et al., 2013) for the waste troubling developing countries is water pollution (Galadima water treatment with the most broadly employed adsorbent et al., 2011). Thousands of these water pollutants are chemical being activated carbon due to simplicity in design, lofty contaminants with many of them of organic origin which adsorption capacity and fast adsorption kinetics (Garba include chlorophenols. Guaiacols are among those chemical et al., 2014). Activated carbons (AC) are the most sought after contaminants with pharmacological properties quite analo- adsorbents (Jodeh et al., 2016) due to their versatile surface gous to those of phenol. Chlorinated guaiacols are closely characteristics, widely utilized for a variety of industrial related to chlorophenols. applications. The conversion of an agricultural waste material into a useful commodity toward the removal of a potential contaminant seems to be an attractive way in economic as well * Corresponding author at: School of Chemical Sciences, Universiti as environmental point of view. Sains Malaysia, 11800 Penang, Malaysia. Tel.: +60 1126116051, +234 8039443335. Optimum conditions for AC preparation from PASH have E-mail address: [email protected] (Z.N. Garba). been reported in our earlier studies but no work has been Peer review under responsibility of University of Bahrain. reported to be done on the adsorption application of the http://dx.doi.org/10.1016/j.jaubas.2015.09.001 1815-3852 Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 18 A.A. Rahim, Z.N. Garba optimal PASH-AC on any type of adsorbate which constitutes pH (2–12) on the 4CG adsorption by PASH-AC and measured the novelty of our work. using a pH meter (Martini instrument, Mi 150). This work is therefore aimed at investigating the effect of Fourier Transform Infrared (FTIR) spectroscopy (FTIR- initial 4CG concentration, adsorption time and pH of the solu- 2000, PerkinElmer) with KBr technique was used in analysing tion on the optimal activated carbon (PASH-AC) using potas- the functional groups on the precursor as well as the PASH- sium oxalate (K2C2O4) as chemical activating agent for the AC surface. The spectra were documented from 4000 to removal of 4-Chloroguiacol (4CG). Kinetic, equilibrium data 400 cm1. modeling as well as thermodynamics study of the spent PASH-AC were also investigated. Similar experiments were 3. Results and discussion carried out under the same conditions with commercial acti- vated carbon for comparison. 3.1. Characterization of PASH-AC

2. Materials and methods The spectra of precursor and PASH-AC in Fig. 1 show an increase and/or decrease of peaks. The broad bands between 2.1. Adsorbate (4-Chloroguiacol) 3500 and 3200 cm1 on the spectra signify the presence of O–H or N–H functional groups, peaks between 3300– Sigma–Aldrich (M) Sdn Bhd, Malaysia supplied the adsorbate 3000 cm1 and 3000–2800 cm1 have been allotted to unsatu- (4CG) used with all the solutions prepared using deionized rated as well as saturated C–H respectively, those bands at water. 4CG has a molecular weight of 158.58 g/mol with chem- 1800–1600 cm1, 1600–1700 cm1 and 1500–1600 cm1 were ical formula of C7H7ClO. for C‚O, C‚C or aromatic rings and C„N respectively (Shi et al., 2010). The bands between 1500 and 1400 cm1 con- 2.2. Preparation of adsorbent (PASH-AC) note the presence of C–C stretching, additionally, the presence of C–O stretching in carboxyl acids, alcohols, phenols and Prosopis africana seed hulls (PASH) used as the precursor was esters was justified by the bands between 1260 and 1 collected from Nigeria. The procedure employed in producing 1050 cm . The weak peak located between 700 and 800 was the PASH-AC was as reported in our erstwhile work (Garba designated to C–OH (out of plane bending) in phenol. As and Afidah, 2014) where the precursor (PASH) was impreg- can be seen from the spectra, numerous functional groups van- ished after carbonization and activation processes. This was nated with the K2C2O4. The optimum preparation conditions applied were as obtained in our previous work (Garba and attributed to thermal degradation effect which resulted in the Afidah, 2014) which produced PASH-AC with reasonable destruction of some intermolecular bonding. yield and significant 4CG removal.

2.3. Removal of 4CG by batch adsorption

Batch adsorption experiments for the 4CG removal by PASH- AC were conducted as reported in our formerly published work (Garba et al., 2014). The 4CG percentage removed at equilibrium (%R) was evaluated as:

C C 4CG removal ð%Þ¼ o e 100 ð1Þ Co where the initial and equilibrium concentrations are denoted as Co and Ce (mg/L), respectively. The adsorbed equilibrium amount of 4CG, qe (mg/g), was estimated by Eq. (2):

ðC C ÞV q ¼ o e ð2Þ e W In order to analyze kinetics of the adsorption process, the 4CG concentration was evaluated at interludes of time. The 4CG amount adsorbed at time t, qt (mg/g) was evaluated using Eq. (3):

ðC C ÞV q ¼ o t ð3Þ t W The pH of the solution was adjusted with 0.1 M HCl and 0.1 M KOH solutions in order to study the effect of initial Figure 1 FT-IR spectra for (a) PASH and (b) PASH-AC. Adsorption of 4-Chloroguiacol from aqueous solution 19

3.2. Influence of adsorption time and 4CG concentrations

The influence of adsorption time on the 4CG removal by PASH-AC for six different adsorbate concentrations at 30 °C is described in Fig. 2. Rapid increase of the 4CG concentra- tions can be observed from the start, with much slower uptake following until equilibrium was established. Equilibrium position was attained at shorter time for lower initial concentrations than at higher initial concentrations as can be observed. The difference in equilibrium time attainment was attributed to the faster extinction or disappearance of adsorbate molecules at different initial concentrations (Hameed et al., 2008). The influence of potassium oxalate activating agent for the development of mesoporous and high surface area of PASH-AC with numerous functional groups as Figure 3 Effect of solution pH on 4CG removal by PASH-AC. seen in characterization results enhanced the faster adsorption process observed.

0 3.3. Effect of solution pH KLQaCe qe ¼ ð4Þ 1 þ KLCe Charge on adsorbent surface, functional group detachment on The isotherm constants associated with adsorption capacity its effective sites, the extent of ionization as well as structural 0 and rate of adsorption were symbolized as Qa (mg/g) and KL changes of adsorbate molecules can be influenced by solution (L/mg) respectively. Eq. (4) was expressed in five different lin- pH. As shown in Fig. 3, 4CG percentage removal shows a sig- ear forms, as tabulated in Table 1, with their major main dis- nificant decrease with an upsurge in the solution pH from 2 to parities connected to the distribution of data as well as the 12. The percentage removal, as high as 95.81% was achieved at parameter determination accuracy (Baccar et al., 2013). pH 2, which was attributed to its high tendency of hydrogen The term describing essential characteristics of the mono- bond formation with the surface of the PASH-AC due to the layer equation is referred to as dimensionless separation factor withdrawing group effect exerted by the methoxy group (RL), defined as (Sadaf et al., 2015): (Hamad et al., 2011). 1 RL ¼ ð5Þ 3.4. Adsorption isotherm modeling 1 þ KLCo

with Co standing for the highest 4CG initial concentration. Three most popular isotherm models (Temkin, Langmuir as Unfavorable adsorption is described by RL > 1, linear if well as Freundlich) were applied to probe the equilibrium data. RL = 1, favorable for 0, (0 < RL < 1) as well as irreversible Langmuir isotherm is one of the highly popular isotherms adsorption if RL =0. for the removal of dyes as well as other organic pollutants Second most widely used isotherm model is Freundlich iso- by adsorption onto activated carbon. The model is explained therm postulated base on surfaces that are heterogeneous. Its by Eq. (4) (Langmuir, 1916): logarithmic form is expressed as (Freundlich, 1906): 1 log q ¼ log K þ log C ð6Þ e F n e

with the two constants symbolized as KF and n measuring the adsorption capacity of the adsorbent as well as how the model

Table 1 Linear forms of Langmuir isotherm. Isotherm Linear form Plot Langmuir-1 1 1 1 1 1 ¼ o þ o vs qe KLQaCe Qa qe Ce Langmuir-2 Ce Ce 1 Ce ¼ o þ o vs Ce qe Qa KLQa qe Langmuir-3 qe o qe qe ¼ þ Qa qevs KLCe Ce Langmuir-4 qe o qe ¼KLqe þ KLQa vs qe Ce Ce o Langmuir-5 1 KLQa 1 1 ¼ KL vs Figure 2 Effect of contact time on 4CG adsorption onto PASH- Ce qe Ce qe AC at various initial concentrations. 20 A.A. Rahim, Z.N. Garba deviates from linearity, respectively. Generally, n > 1 suggests five linear equations were not the same as can be observed favorable adsorption of adsorbate on the adsorbent. The from Table 2, because the transformations change the original greater the value of n, the more sturdy the adsorption strength. error distribution (Baccar et al., 2013). Based on the R2 values, Temkin model was based on how indirect adsorbent/adsor- the best fit should have been Langmuir-1 or Langmuir-5 iso- bate interactions influence the adsorption isotherms. Its linear therms in comparison with the other isotherm equations form is expressed as (Temkin and Pyzhev, 1940): because they showed the largest values (R2 = 0.9986). 2 RT RT But according to Baccar et al. (2013) the highest R does qe ¼ ln A þ lnCe ð7Þ not necessarily describe the most superlative transformation. b b So as observed from Table 2, Langmuir-1 and 5 had the high- RT 2 2 2 where b = B (J/mol) and A (L/g) are Temkin constants, est R values (R = 0.9986) but their v values (1.130) were which are related to heat of sorption and maximum binding also larger, higher than Langmuir-2 (0.234), therefore they energy, respectively, R is the gas constant (8.31 J/mol K) and cannot be concluded to perfectly describe the equilibrium data. T (K) is the absolute temperature. It can also be seen from Table 2 that v2 value of the Langmuir- To compare the validity of each model, chi square (v2) was 2 isotherm (0.234) was lower than those obtained from the Fre- incorporated since correlation coefficient (R2) may not justify undlich (0.759) and Temkin (59.859), therefore, the maximum the basis for selecting the best adsorption model because it 0 ; adsorption capacity (QaÞ sorption energy (KL) and separation only signifies the fit between linear forms of the isotherm equa- factor (RL) values of 498.67 mg/g, 0.038 and 0.069, respectively tions and experimental data and while the suitability between were adopted from the Langmuir-2 equation. experimental and predicted values of the adsorption capacity The high Q0 of 498.67 mg/g observed in this study was v2 v2 a is described by chi square ( ). The lower the value, the bet- attributed to the relatively high surface area of the PASH- ter the fit. AC and its mesoporous structure (Garba and Afidah, 2014). Table 2 summarizes the parameters obtained from the 2 It compares well with those obtained from the literature as adsorption isotherm models applied with their respective R summarized in Table 3. and v2 values. The Langmuir parameters obtained from the 3.5. Adsorption kinetic studies

The kinetics of 4CG adsorption was investigated by applying Table 2 Langmuir (1–5), Freundlich and Temkin isotherm Lagergren pseudo-first order and pseudo-second order (1 and model parameter correlation coefficients and chi square values 2) models. The pseudo-first-order linear equation was given for 4CG adsorption on PASH-AC at 30 °C. as (Lagergren and Svenska, 1898): k Isotherm Parameters q q q 1 t logð e tÞ¼log e : ð8Þ 0 R R2 v2 2 303 Qa (mg/g) L where k is the pseudo-first-order rate constant (h1). Langmuir 1 Langmuir-1 407.29 0.054 0.9986 1.130 The two linear forms of pseudo-second-order equations Langmuir-2 498.67 0.069 0.9957 0.234 were expressed as Eqs. (9) and (10): Langmuir-3 481.57 0.067 0.9792 8.728 t 1 1 ¼ þ t ð9Þ Langmuir-4 493.09 0.055 0.9792 1.420 q k q2 q Langmuir-5 413.59 0.055 0.9986 2.860 t 2 e e  K (mg/g (L/mg) nR2 v2 1 1 1 1 F 10 ¼ 2 þ ð Þ Freundlich 2.3276 1.382 0.9904 0.759 qt k2qe t qe 2 2 A (L/g) B (J/mol) R v where k2 (g/mgh) is the pseudo-second-order rate constant. 2 v2 Temkin 0.644 85.518 0.9470 59.859 The values of qe, k1, R and obtained after the linear 2 2 plots of Eq. (8) and qe, k2, R and v from the plots of Eqs.

Table 3 Comparison of maximum monolayer adsorption capacity of various CPs on different adsorbents.

0 Adsorbent Adsorbate Qa (mg/g) References PASH-AC 4-Chloroguaiacol 498.67 This work Commercial activated carbon 4-Chloroguaiacol 276.88 This study Oil palm shell activated carbon 4-Chloroguaiacol 454.45 Hamad et al. (2010) Oil palm shell activated carbon 4-chloro2-methoxy phenol 323.62 Hamad et al. (2011) Rattan sawdust based activated carbon 4-chlorophenol 188.68 Hameed et al. (2008) Cattail fibre-based activated carbon 2,4-Dichlorophenol 142.86 Ren et al. (2011) Rice straw carbon 3-chlorophenol 14.2 Wang et al. (2007) Adsorption of 4-Chloroguiacol from aqueous solution 21

Table 4 Pseudo-first-order and pseudo-second-order (1 and 2) kinetic model parameters of 4CG adsorption on PASH-AC at 30 °C.

Co (mg/L) qe,exp (mg/g) Pseudo-first-order Pseudo-second-order-1 Pseudo-second-order-2 2 2 2 2 2 2 k1 (1/h) qe,cal (mg/g) R v k2 (g/mg h) qe,cal (mg/g) R v k2 (g/mg h) qe,cal (mg/g) R v 30 28.51 0.330 14.73 0.981 0.234 0.100 26.88 0.996 0.003 0.178 24.57 0.960 0.019 60 56.60 0.482 41.21 0.989 0.074 0.023 59.52 0.997 0.003 0.029 55.87 0.998 0.001 100 93.81 0.377 57.73 0.988 0.148 0.021 90.91 0.992 0.001 0.037 81.97 0.969 0.016 150 139.31 0.430 89.78 0.997 0.126 0.013 138.89 0.993 0.009 0.026 120.48 0.958 0.018 250 228.38 0.356 134.15 0.978 0.170 0.010 217.39 0.995 0.002 0.016 200.00 0.972 0.015 350 307.86 0.421 215.48 0.990 0.090 0.005 312.50 0.996 0.002 0.0064 285.71 0.991 0.005

of these kinetic models in describing the 4CG adsorption pro- cess, chi-square (v2) statistical analysis was also applied. As can be observed from Table 4, the v2 values acquired from both the pseudo-second-order 1 and 2 models (0.001–0.019) were lower than those obtained from the pseudo-first-order (0.074–0.234) which further confirms the pseudo-second- order equation to be the most preeminent kinetic model in describing the 4CG adsorption onto PASH-AC. The kinetic models were limited in terms of identifying the diffusion mechanisms as well as the rate controlling steps in the adsorption process, as result of that limitation; intraparticle diffusion model was further applied. The intraparticle diffu- sion equation is expressed as:

1=2 Figure 4 Plot of intraparticle diffusion model for adsorption of qt ¼ kipt þ C ð11Þ 4CG onto PASH-AC at 30 °C. where kip is rate constant of the intra-particle diffusion equa- tion and C gives information about the boundary layer thick- ness: larger value of C is associated with the boundary layer 1/2 (9) and (10) (figures not shown) for the 4CG adsorption on the diffusion effect. When the linear plot qt versus t is linear PASH-AC are reported in Table 4. and passes through the origin, it connotes that the adsorption As can be observed from Table 4, the trends of R2 values process follows the intraparticle diffusion model, and that the (0.981–0.997) for the pseudo-first-order model were not coher- only rate limiting step involved is intraparticle diffusion, if not, ent. Also, the calculated and experimental qe values were not in then it connotes that intraparticle diffusion is not the only rate good concord with each other, indicating the inappropriate- limiting step involved (Fan et al., 2011). ness of pseudo first-order kinetic model in describing 4CG The intraparticle diffusion plots for the adsorption of 4CG adsorption onto PASH-AC. However, the harmony between on PASH-AC at 30 °C are depicted by Fig. 4. experimental and calculated qe values obtained from pseudo- As can be seen from Fig. 4, a very rapid adsorption was second-order (1 and 2) models were lofty with all the R2 values described by the first sharper region completed attributed to obtained very near to unity, confirming that the most suitable a strong electrostatic attraction between 4CG and the model to describe 4CG adsorption onto PASH-AC was PASH-AC external surface. The next stage describes a steady pseudo-second-order model. To further confirm the suitability adsorption stage, which can be attributed to intraparticle

Table 5 Intraparticle diffusion model parameters for the adsorption of 4CG onto PASH-AC.

Co (mg/L) Intraparticle diffusion model 1/2 1/2 2 2 kp2 (mg/g h ) kp3 (mg/g h ) C2 C3 (R2) (R3) 30 6.847 – 11.007 25.750 0.9561 – 60 20.593 0.2227 9.692 52.633 0.9534 0.4919 100 26.770 0.5229 27.354 87.648 0.9660 0.4919 150 42.560 0.2742 37.794 135.630 0.9835 0.4919 250 64.115 – 69.124 – 0.9498 – 350 106.980 – 55.924 – 0.9639 – 22 A.A. Rahim, Z.N. Garba

upsurge in both adsorption time as well as initial 4CG concen- Table 6 Thermodynamic parameters for the adsorption of tration. The adsorption process was more promising in lower 4CG onto PASH-AC at different temperatures. pH solution with the Langmuir-2 model being the most appro- DH (kJ/mol) DS (J/mol K) DG (kJ/mol) priate in describing the equilibrium data. The kinetics data 303 K 313 K 323 K obeyed pseudo-second-order model. 4CG adsorption onto PASH-AC was primarily presided by particle diffusion 1.49 24.30 5.94 5.96 6.43 according to the Boyd plot. The positive DH values observed connoted the adsorption process to be endothermic. The adsorption potential of the PASH-AC competed satisfactorily diffusion of the 4CG molecule through the activated carbon’s with earlier studied adsorbents. Based on the obtained results, pores. Third stage exists in few cases, especially when the 4CG the PASH-AC produced can be used effectively to tackle pol- initial concentrations are high. At that stage, the intraparticle lution problems posed by chloroguaicols in the environment. diffusion starts to slow down (Wang et al., 2010). 2 The values of kpi, Ci and R obtained are given in Table 5. Acknowledgement The values of kp2 as can be seen from Table 5 increased with upsurge in the initial 4CG concentration, which was attributed Research University Grant 1001/PKIMIA/854002 from to the greater driving force. The values of C2 and C3 also Universiti Sains Malaysia that ensued in this article was recog- increased with the increase in 4CG concentration from 30 to nized by the authors. 350 mg/L signifying an increase in the thickness of the bound- ary layer (Khaled et al., 2009). References In the second and third stages as can be observed from Fig. 4, the linear lines did not pass through the origin which Baccar, R., Bla´ nquez, P., Bouzid, J., Feki, M., Attiya, H., Sarra` , M., suggested the presence of intraparticle diffusion along with 2013. Modeling of adsorption isotherms and kinetics of a tannery possibility of involvement of some other rate controlling steps dye onto an activated carbon prepared from an agricultural by- in the adsorption process (Maksin et al., 2012). product. Fuel Process. Technol. 106, 408–415. El Haddad, M., Mamouni, R., Saffaj, N., Lazar, S., 2012. Removal of 3.6. Adsorption thermodynamic studies a cationic dye – Basic Red 12 – from aqueous solution by adsorption onto animal bone meal. J. Assoc. Arab Univ. Basic Appl. Sci. 12, 48–54. D D Gibb’s free energy change ( G), enthalpy change ( H) and El Haddad, M., Slimani, R., Mamouni, R., ElAntri, S., Lazar, S., entropy change (DS) are the most popular thermodynamic 2013. Removal of two textile dyes from aqueous solutions parameters that were considered and studied in this work. Van’t onto calcined bones. J. Assoc. Arab Univ. Basic Appl. Sci. 14, Hoff equation was employed in determining the thermodynamic 51–59. parameters which was expressed as (Slimani et al., 2014): Fan, J., Zhang, J., Zhang, C., Ren, L., Shi, Q., 2011. Adsorption of 2,4,6-trichlorophenol from aqueous solution onto activated carbon DS DH ln KD ¼ ð12Þ derived from loosestrife. Desalination 267, 139–146. R RT Freundlich, H.M.F., 1906. Over the adsorption in solution. J. Phys. where R (8.314 kJ/mol) is the universal gas constant; T (K) is Chem. 57, 385–470. qe Galadima, A., Garba, Z.N., Leke, L., Almustapha, M.N., Adam, I.K., the absolute temperature; KD ¼ is the distribution coeffi- Ce 2011. Domestic water pollution among local communities in cient; qe (mg/g) is the amount of adsorbate adsorbed on the nigeria––causes and consequences. Eur. J. Sci. Res. 52 (4). sorbent per unit mass. A linear plot of lnK against 1/T gives D Garba, Z.N., Afidah, A.R., 2014. Process optimization of K2C2O4- a graph (Fig. not shown) with DH and DS obtained from the activated carbon from Prosopis africana seed hulls using response slope and intercept respectively. DG was evaluated from the surface methodology. J. Anal. Appl. Pyrol. 107, 306–312. relation below: Garba, Z.N., Afidah, A.R., Hamza, S.A., 2014. Potential of borassus aethiopum shells as precursor for activated carbon preparation by DG RT ln K 13 ¼ D ð Þ physico-chemical activation; optimization, equilibrium and kinetic The thermodynamic parameters obtained for the adsorp- studies. J. Environ. Chem. Eng. 2, 1423–1433. tion of the 4CG on PASH-AC at three different temperatures Hamad, B.K., Noor, A.M., Afida, A.R., Mohd Asri, M.N., 2010. High are reported in Table 6. removal of 4-chloroguaiacol by high surface area of oil palm shell- activated carbon activated with NaOH from aqueous solution. Positive values were obtained for both DH and DS, implying Desalination 257 (1–3), 1–7. that the 4CG adsorption process was endothermic with random Hamad, B.K., Ahmad, M.N., Afidah, A.R., 2011. Removal of 4- characteristics. The Gibb’s free energy of change was sponta- chloro-2-methoxy phenol by adsorption from aqueous solution D neous as can be seen by the negative values obtained. The G using oil palm shell carbon activated by K2CO3. J. Phys. Sci. 22, values also confirmed the adsorption of 4CG onto PASH-AC 41–58. to be a physical process with the physical adsorption values Hameed, B.H., Chin, L.H., Rengaraj, S., 2008. Adsorption of 4- ranging from 20 to 0 kJ/mol while value from 80 to chlorophenol onto activated carbon prepared from rattan sawdust. 400 kJ/mol describes chemical adsorption (Li et al., 2010). Desalination 225 (1–3), 185–198. Jodeh, S., Abdelwahab, F., Jaradat, N., Warad, I., Jodeh, W., 2016. Adsorption of diclofenac from aqueous solution using Cyclamen 4. Conclusions persicum tubers based activated carbon (CTAC). J. Assoc. Arab Univ. Basic Appl. Sci. 20, 32–38. PASH-AC was produced from Prosopis africana seed hulls. Its Khaled, A., El-Nemr, A., El-Sikaily, A., Abdelwahab, O., 2009. adsorption capacity was observed to be increasing with an Removal of Direct N Blue106 from artificial textile dye effluent Adsorption of 4-Chloroguiacol from aqueous solution 23

using activated carbon from orange peel: adsorption isotherm and Ren, L., Zhang, J., Li, Y., Zhang, C., 2011. Preparation and evaluation kinetic studies. J. Hazard. Mater. 165, 100–110. of cattail fiber-based activated carbon for 2,4-dichlorophenol and Lagergren, S., Svenska, B.K., 1898. On the theory of so-called 2,4,6-trichlorophenol removal. Chem. Eng. J. 168, 553–561. adsorption of dissolved substances. R. Swed. Acad. Sci. Doc. 24, Sadaf, S., Bhatti, H.N., Nausheen, S., Amin, M., 2015. Removal of Cr 1–13. (VI) from wastewater using acid-washed zero-valent iron catalyzed Langmuir, I., 1916. The constitution and fundamental properties of by polyoxometalate under acid conditions: efficacy, reaction solids and liquids part I solids. J Am. Chem. Soc. 38, mechanism and influencing factors. J. Taiwan Inst. Chem. Eng. 2221–2295. 47, 160–170. Li, Q., Yue, Q., Su, Y., Gao, B., Sun, H., 2010. Equilibrium, Shi, Q.Q., Zhang, J., Zhang, C.L., Li, C., Zhang, B., Hu, W.W., Xu, J. thermodynamics and process design to minimize adsorbent amount T., 2010. Preparation of activated carbon from cattail and its for the adsorption of acid dyes onto cationic polymer-loaded application for dyes removal. J. Environ. Sci. 22, 91–97. bentonite. Chem. Eng. J. 158, 489–497. Slimani, R., El Ouahabi, I., Abidi, F., El Haddad, M., Regti, A., Maksin, D.D., Nastasovic´ , A.B., Milutinovic´ -Nikolic´ , A.D., Surucˇ ic´ , Laamari, M., El Antri, S., Lazar, S., 2014. Calcined eggshells as a L.T., Sandic´ , Z.P., Hercigonja, R.V., Onjia, A.E., 2012. Equilib- new biosorbent to remove basic dye from aqueous solutions: rium and kinetics study on hexavalent chromium adsorption onto thermodynamics, kinetics, isotherms and error analysis. J. Taiwan diethylene triamine grafted glycidyl methacrylate based copoly- Inst. Chem. Eng. 45, 1578–1587. mers. J. Hazard. Mater. 209, 99–110. Temkin, M.J., Pyzhev, V., 1940. Recent modifications to Langmuir Noreen, S., Bhatti, H.N., 2014. Fitting of equilibrium and kinetic data isotherms. Acta Physicochim. 12, 217–222. for the removal of Novacron Orange P-2R by sugarcane bagasse. J. Wang, S.L., Tzou, Y.M., Lu, Y.H., Sheng, G., 2007. Removal of 3- Ind. Eng. Chem. 20, 1684–1692. chlorophenol from water using rice-straw-based carbon. J. Hazard. Noreen, S., Bhatti, H.N., Nausheen, S., Sadaf, S., Ashfaq, M., 2013. Mater. 147, 313–318. Batch and fixed bed adsorption study for the removal of Drimarine Wang, L., Zhang, J., Zhao, R., Li, C., Li, Y., Zhang, C., 2010. Black CL-B dye from aqueous solution using a lignocellulosic Adsorption of basic dyes on activated carbon prepared from waste: a cost affective adsorbent. Ind. Crops Prod. 50, Polygonum orientale Linn: equilibrium, kinetic and thermody- 568–579. namic studies. Desalination 254, 68–74. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016) 21,24–30

University of Bahrain Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com

ORIGINAL ARTICLE A thermodynamical, electrochemical and surface investigation of Bis (indolyl) methanes as Green corrosion inhibitors for mild steel in 1 M hydrochloric acid solution

Chandrabhan Verma, Pooja Singh, M.A. Quraishi *

Department of Chemistry, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India

Received 23 February 2015; revised 20 April 2015; accepted 25 April 2015 Available online 23 May 2015

KEYWORDS Abstract The influence of three Bis (indolyl) methanes (BIMs) namely, 3,30-((4-nitrophenyl) methy- 0 Mild steel; lene) bis (1H-indole) (BIM-1), 3,3 -(phenyl methylene) bis (1H-indole) (BIM-2) and 4-((1H-indol-2- Corrosion; yl)(1H-indol-3-yl) methyl) phenol (BIM-3) on the mild steel corrosion in 1 M HCl was studied by EIS; weight loss, electrochemical, scanning electron microscopy (SEM), and dispersive X-ray spec- Tafel polarization; troscopy (EDX) methods. Results showed that BIM-3 shows maximum inhibition efficiency of SEM/EDX 98.06% at 200 mg L1 concentration. Polarization study revealed that the BIMs act as mixed type inhibitors. Adsorption of BIMs on the mild steel surface obeyed the Langmuir adsorption isotherm. The weight loss and electrochemical results were well supported by SEM and EDX studies. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction procedure, high yield, high selectivity and clean reaction (Joshi et al., 2010). Organic compounds particularly, N-heterocyclic have been Indole and its derivatives have received considerable atten- reported as effective corrosion inhibitors for mild steel against tion of synthetic chemists due to their several biological appli- corrosion during several industrial processes (Solmaz, 2014; cations such as antibacterial, cytotoxic, antioxidative, Musa et al., 2012; Mahdavian and Ashhari, 2010; Ozkir insecticidal activities and bioactive metabolites of terrestrial et al., 2012). Ultrasound irradiation has immerged as a power- and marine origin (Surasani et al., 2013). In our present inves- ful technique for the synthesis of various heterocyclic com- tigation we have synthesized and studied the corrosion inhibi- pounds of industrial and biological interest (Goharshad tion efficiency of three Bis (indolyl) methanes on mild steel et al., 2009) due to their shorter reaction time, simple operating corrosion in 1 M HCl. The criteria behind selecting these com- pounds as corrosion inhibitors were that: (a) they can be easily * Corresponding author. Tel.: +91 9307025126; fax: +91 542 synthesized from commercially available and relatively cheap 2368428. starting materials (b) contain –OH, –NO2 and hetero- E-mail addresses: [email protected], maquraishi@ aromatic rings through which they can adsorb and inhibit cor- rediffmail.com (M.A. Quraishi). rosion (c) they were effective even at low concentration and (d) Peer review under responsibility of University of Bahrain. they were highly soluble in testing medium. Previously, few http://dx.doi.org/10.1016/j.jaubas.2015.04.003 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Investigation of Bis (indolyl) methanes as Green corrosion inhibitors 25 authors reported the corrosion inhibition efficiency of the 2.4. Electrochemical experiments indole and its derivatives in acid solution for different metals (Norr, 2008; Popova and Christov, 2006; Lowmunkhong As described earlier (Verma et al., 2014a), a typical three elec- et al., 2010; Quartarone et al., 2008). trodes glass cell consisting of a highly pure platinum mesh as counter electrode, a saturated calomel as reference electrode 2. Experimental section and mild steel specimen as working electrode was used for elec- trochemical studies. The Tafel and EIS measurements were 2.1. Material carried out using a Gamry Potentiostat/Galvanostat (Model G-300) with EIS Software Gamry Instruments Inc., USA. The mild steel specimens having composition (wt.%): Echem Analyst 5.0 Software package was applied to analyze C = 0.076, Mn = 0.192, P = 0.012, Si = 0.026, Cr = 0.050, the electrochemical data. The cathodic and anodic Tafel slopes Al = 0.023, and remainder Fe were used in present study. were recorded by changing the electrode potential inevitably The test solution (1 M HCl) was prepared by dilution of ana- from 0.25 to +0.25 V vs. corrosion potential (Ecorr) at a con- 1 lytical grade HCl (MERK, 37%) in double deionized water. stant sweep rate of 1.0 mV s . The EIS studies were carried out under potentiostatic condition in a frequency range of 2.2. Synthesis of inhibitors (BIMs) 100 kHz–0.01 Hz. The amplitude of the AC sinusoid wave was 10 mV. All the Tafel and EIS studies were performed in naturally aerated solution of 1 M HCl in the absence and the In the present study Bis (indolyl) methanes (BIMs) were synthe- presence of 200 mg L1 concentration of BIMs after 30 min sized as described earlier (Sonar et al., 2009). The synthetic rout immersion time. for BIMs is shown in Scheme 1. The purity of products was determined by TLC method. The characterization data of the 2.5. SEM/EDX analysis synthesized compounds are as follows: BIM-1 (3,30-((4- nitrophenyl) methylene) bis (1H-indole, –R = –Ph (4-NO2))): MP: 223-224 C, IR (KBr, cm1): 3428, 2829, 2245, 1680–1660, The SEM model Ziess Evo 50XVP instrument was used for the 1522, 1230, 845, 739, 641. BIM-2 (3,30-(phenyl methylene) bis mild steel surface analysis with and without BIMs using accel- (1H-indole) –R = Ph): MP: 125–127 C, IR (KBr, cm1): erating voltage of 50 kV at 500· magnifications. Before SEM 3465, 2811, 2275, 1482, 1130, 825, 758, 611. BIM-3 (4-((1H- and EDX analysis the mild steel samples were immersed for indol-2-yl)(1H-indol-3-yl) methyl) phenol, –R = Ph(4-OH)): 3 h in the absence and the presence of BIMs. The elemental MP: 123–125 C, IR (KBr, cm1): 3623, 3475, 2831, 2356, composition was determined using energy dispersive X-ray 1453, 1238, 845, 734, 623. spectroscopy (EDX) coupled with SEM.

2.3. Gravimetric experiment 3. Result and discussion

The weight loss experiments in the absence and the presence of 3.1. Weight loss measurements different concentrations of BIMs were carried out to optimize the concentration of BIMs as described earlier (Verma et al., 3.1.1. Effect of concentration 2014). The corrosion rate (CR), percentage inhibition efficiency Variation of the inhibition efficiency (g%) at different studied (g%) and surface coverage (h) were calculated using following concentrations of BIMs is shown in Fig. 1. It is obvious that equations. 87:6W C ¼ ð1Þ R Atd C C g% ¼ R RðiÞ 100 ð2Þ CR C C h ¼ R RðiÞ ð3Þ CR where, W is the weight loss in mg, A is the area (cm2) of the mild steel sample exposed to 1 M HCl, t is the immersion time 3 (3 h), d is the density of mild steel (g cm ) and CR and CR(i) are the corrosion rates in the presence and the absence of BIMs, respectively.

H N

H N O Alum (10%) + 2 Solvenr free, US R R H

H NH Figure 1 Variation of inhibition efficiency with BIMs concen- BIM-1: R = -Ph(4-NO2), BIM-2: R = -Ph, BIM-3: R = -Ph (4-OH) tration of mild steel immersed in 1 M HCl obtained by weight loss Scheme 1 Synthetic route for investigated BIMs. measurement. 26 C. Verma et al. the values of g% increases on increasing BIMs concentration. 3.1.2. Effect of temperature 1 The maximum g% was obtained at 200 mg L concentration To investigate the effect of temperature on inhibition perfor- further increase in concentration does not cause any significant mance of BIMs, the weight loss experiments were also per- change in the inhibition performance suggesting that formed at different temperatures (308–338 K). The values of 200 mg L1 is the optimum concentration. The increase in the CR at different studied temperatures are listed in Table 1. BIMs concentration increases the surface coverage (h) through It is apparent from results that the value of the CR increases adsorbing on its surface and therefore, increases inhibition effi- on increasing temperature for inhibited as well as uninhibited ciency (Yadav et al., 2013). solutions. This increased values of CR is attributed to desorp- tion of the adsorb BIMs molecules from mild steel surface at elevated temperatures, resulting in enhanced CR (Barmatov et al., 2015). Table 1 Variation of corrosion rate with temperature in the The temperature dependency of corrosion rate can be best absence and presence of optimum concentration of BIMs. represented by the Arrhenius and transition state equations

2 1 (Deng et al., 2011): Temperature (K) Corrosion rate (CR) (mg cm h ) E Blank BIM-1 BIM-2 BIM-3 logðC Þ¼ a þ log k ð4Þ R 2:303RT 308 7.60 0.46 0.30 0.13   318 11.0 1.23 1.10 0.83 RT DS DH 328 14.3 2.03 1.80 1.56 CR ¼ exp exp ð5Þ 338 18.6 3.30 2.76 2.46 Nh R RT

where, h is Plank’s constant, Ea is the apparent activation energy, N is Avogadro’s number, DS* is the entropy of activation and DH* is the enthalpy of activation R is the gas constant, T is the temperature, k is the Arrhenius pre- 2.4 exponential factor. The values of activation parameter (Ea) and transition state Parameters (DH*, DS*) were calculated 2.1 from Arrhenius plots (Fig. 2a) and transition state plots ) 1 -

h 1.8 2 - 1.5 Table 2 Activation parameters for mild steel dissolution in 1 M HCl in the absence and presence of optimum concentra- (mg cm 1.2 R tion of BIMs. C 0.9 1 * 1 * Blank Inhibitor Ea (kJ mol ) DH (kJ mol ) DS (J/mol K) log BIM-1 0.6 Blank 28.48 26.04 148.9 BIM-2 BIM-1 62.72 54.06 147.12 BIM-3 0.3 BIM-2 65.83 54.53 127.5 BIM-3 73.93 72.43 68.2 2..95 3..00 3..05 3.10 3.15 3.20 3.25 [(1//T)103]K-1 (a) 1.8 BIM-1 1.6 BIM-2 BIM-3 1.4

) 1.2 θ

/1- 1.0 θ 0.8 log ( 0.6 0.4 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 log C (b) (inh)

Figure 2 Arrhenius plots for mild steel in 1 M HCl in the Figure 3 Langmuir isotherm plots for mild steel in 1 M HCl absence and presence of different concentrations of BIMs. solution containing different concentration of BIMs. Investigation of Bis (indolyl) methanes as Green corrosion inhibitors 27

Table 3 The values of Kads and DGads for mild steel in the absence and presence of optimum concentration of BIMs in 1 M HCl at different studied temperatures.

4 1 1 Inhibitor Kads (10 M ) DGads (kJ mol ) 308 318 328 338 308 318 328 338 BIM-1 18.43 5.37 3.54 2.37 35.75 34.39 33.53 33.38 BIM-2 20.98 5.87 3.89 2.46 35.98 34.13 33.23 33.35 BIM-3 25.85 6.35 4.65 2.85 36.42 35.52 34.43 33.47

(Fig. 2b), respectively and given in Table 2. The increased isotherms such as Langmuir, Temkin, Freundluich, value of the Ea in the presence of BIMs is attributed due to BockrisSwinkles and Flory–Huggins isotherms were tested. physical adsorption that takes place during first step of adsorp- tion processes (Faustin et al., 2015). The positive values of DH* reflect the endothermic nature of mild steel dissolution in acidic medium and the negative values of DS* suggest the for- mation of activated complex in the rate determining step which represent dissociation rather than association suggesting that disorderness increases on going from reactant to activated complex (Faustin et al., 2015).

3.1.3. Adsorption isotherm and energy of adsorption In order to gain some mechanistic information about adsorption of BIMs on mild steel surface, several adsorption

Figure 4 Tafel polarization curves for mild steel obtained in 1 M HCl containing different concentrations BIMs.

Table 4 Tafel Polarization parameters for mild steel in 1 M HCl solution in the absence and presence of optimum concen- tration of BIMs.

Inhibitor Ecorr icorr ba bc g% (V/SCE) (lA/cm) (mV/dec) (mV/dec) Blank 445 1150 70.5 114.6 – BIM-1 466 95.8 66.8 151.0 91.66 BIM-2 463 94.1 61.3 138.4 91.82 Figure 5 (a) Nyquist plots for mild steel obtained in 1 M HCl BIM-3 527 63.51 80.9 127.9 94.47 containing different concentrations BIMs. (b) Equivalent circuit used to fit the EIS data for mild steel in 1 M HCl. 28 C. Verma et al.

The surface coverage values (h) as a function of logarithm of given in Table 3. The values of Kads related to the free energy BIMs concentration were tested graphically to obtain the best of adsorption (DGads) by following relation (Bahrami et al., adsorption isotherm. In our present study the Langmuir iso- 2010): therm gave the best fit which can be best represented by follow- ing equation (Verma et al., 2014b): DGads ¼ –RT ln ð55:5KadsÞð7Þ

CðinhÞ 1 ¼ þ CðinhÞ ð6Þ The value 55.5 in above equation represents the concentra- h K ðadsÞ tion of water in acid solution in mol L1. Generally, a higher where, C(inh) is the inhibitor concentration and Kads is the equi- value of Kads associated with higher tendency to adsorb on librium constant for the adsorption–desorption process. The mild steel surface. In our present study the Kads value of differ- values of Kads were calculated at different studied temperatures ent BIMs follows the order: BIM-3 > BIM-2 > BIM-1 which from the intercept of the Langmuir plot shown in Fig. 3 and is in accordance with the order of the g%. In our present

Table 5 Electrochemical impedance parameters obtained from EIS measurements for mild steel in 1 M HCl in the absence and presence of optimum concentration of BIMs.

2 2 2 Inhibitor Rs (X cm ) Rct (X cm ) Cdl (lFcm ) n g% Goodness of fit Blank 1.12 9.58 106.21 0.827 – 3.735 · 103 BIM-1 0.73 161.8 36.68 0.827 94.07 1.332 · 103 BIM-2 1.11 199.9 29.99 0.848 95.20 674.1 · 106 BIM-3 1.05 264.4 24.80 0.854 97.44 858.9 · 106

Figure 6 SEM micrographs of mild steel surfaces in the absence (a) and presence of optimum concentration of BIM-1 (b), BIM-2 (c) and BIM-3 (d). Investigation of Bis (indolyl) methanes as Green corrosion inhibitors 29 investigation values of DGads vary from 33.38 to gives one time constant and shows a single maximum which is 36.42 kJ mol1 (Table 3) that signifying the both physical a characteristic response of mild steel corrosion in acid solu- and chemical i.e. physiochemisorption of BIMs (Daoud tion. It can be observed from the Bode plot that phase angle et al., 2015). The negative sign of DGads ensures the spontane- significantly increased in the presence of BIMs due to the for- ity of the adsorption process (Daoud et al., 2015). mation of the protecting film by them on the mild steel surface (Verma et al., 2014). This finding suggests that adsorption of 3.2. Electrochemical measurements the BIMs on mild steel surface decreases the surface roughness and therefore, increases the phase angle values. 3.2.1. Polarization study Fig. 4 represents the Tafel polarization curves for mild steel 3.3. Surface investigation corrosion in 1 M HCl in the absence and the presence of opti- mum concentration of BIMs at 308 K. The polarization 3.3.1. SEM analysis parameters namely corrosion potential (Ecorr), the corrosion The SEM micrographs of mild steel surface after 3 h immer- current density (icorr), anodic Tafel slope (ba), cathodic Tafel sion are shown in Fig. 6. The surface morphology in the slope (bc) and corresponding g% were calculated from Tafel absence of BIMs (Fig. 6a) is damaged surface. However, in extrapolation method and given in Table 4. The g% was calcu- the presence of BIMs (Fig. 6b–d) the surface of mild steely sig- lated using following equation: nificantly improved. This finding further supports that BIMs

0 i inhibit the mild steel corrosion by adsorption mechanism. icorr icorr g% ¼ 0 100 ð8Þ icorr 3.3.2. EDX analysis 0 i where, icorr and icorr are the corrosion current densities in The EDX spectra in the absence and the presence of the opti- the absence and the presence of the BIMs, respectively. It mum concentration of the BIMs after 3 h immersion are can be observed that addition of the BIMs significantly shown in Fig. 7. The EDX spectrum without BIMs gives sig- decreases the values of icorr suggesting that BIMs strongly nals only for carbon (C) and Fe as shown in Fig. 7a. adsorb on the mild steel surface in 1 M HCl and retard However, the EDX spectra of mild steel in the presence of the electrochemical reaction occurring on metal surface due BIMs (Fig. 7b–d) showed characteristics signal for N and O to the presence of protective film (Ansari and Quraishi, along with C and Fe. The presence of N and O in the EDX 2015). From Fig. 4 it can also be observed that both catho- spectra suggests that BIMs inhibit mild steel corrosion by dic and anodic reactions were affected in the presence of adsorbing on the mild steel surface. Moreover, intensity of N BIMs. However, cathodic reactions are comparatively more and O increases in the order: BIM-3 > BIM-2 > BIM-1 affected than anodic reactions without causing any signifi- cant change in Ecorr values (<85 mV) suggesting that BIMs are mixed type but predominantly cathodic inhibitors (Anejjar et al., 2014).

3.2.2. EIS measurements Fig. 5a represents the typical Nyquist and Bode plots in the absence and the presence of optimum concentration of the BIMs. The EIS parameters namely solution resistance (Rs), charge transfer resistance (Rct), double layer capacitance (Cdl), and corresponding g% were calculated from EIS mea- surements using equivalent circuit (Fig. 5b) and given in Table 5. The increased values of Rct suggest that presence of BIMs creates a barrier for mass and charge transfer pro- cess (Eddy et al., 2014). The double layer capacitance (Cdl) in the present study was calculated using the following relation: Yxn1 C ¼ ð9Þ dl sinðnðp=2ÞÞ where Y is the amplitude comparable to a capacitance (with a lFcm2), x is the angular frequency, n is the phase shift, which is a measure of surface roughness. Generally, the high value of n is associated with high surface coverage. From Table 5, it is observed that values of Cdl decrease in the pres- ence of BIMs which can be attributed due to decrease in the local dielectric constant and/or an increase in the thickness of electric double layer (Bammou et al., 2014). Fig. 5c represents the Bode impedance magnitude and Figure 7 EDX spectra of mild steel surfaces in the absence (a) phase angle plot for mild steel in the absence and the presence and presence of optimum concentration of BIM-1 (b), BIM-2 (c) of 200 gm L1 concentration of BIMs. As seen, the Bode plot and BIM-3 (d). 30 C. Verma et al. suggesting that BIM-3 has the strongest tendency to adsorb on Geissospermum leave in 1 M hydrochloric acid: electrochemical the mild steel surface among the studied BIMs. and phytochemical studies. Corros. Sci. 92, 287–300. Goharshad, E.K., Ding, Y., Jorabchi, M.N., Nancarrow, P., 2009. Ultrasound-assisted green synthesis of nanocrystalline ZnO in 4. Conclusion the ionic liquid [hmim][NTf2]. Ultrason. Sonochem. 16, 120– 123. Results show that BIMs act as good corrosion inhibitors for Joshi, R.S., Mandhane, P.G., Diwakar, S.D., Gill, C.H., 2010. mild steel in 1 M HCl and their inhibition efficiency increases Ultrasound assisted green synthesis of bis(indol-3-yl)methanes catalyzed by 1-hexenesulphonic acid sodium salt. Ultrason. with increasing concentration. The negative value of DGads suggests that BIMs adsorb spontaneously and their adsorption Sonochem. 17, 298–300. obeys the Langmuir adsorption isotherm. The polarization Lowmunkhong, P., Ungthararak, D., Sutthivaiyakit, P., 2010. Tryptamine as a corrosion inhibitor of mild steel in hydrochloric study revealed that the BIMs act as mixed type inhibitors. acid solution. Corros. Sci. 52, 30–36. The presence of the BIMs increases the charge transfer values Mahdavian, M., Ashhari, S., 2010. Corrosion inhibition performance and therefore inhibits mild steel corrosion. The SEM and EDX of mercaptobenzimidazole and 2-mercaptobenzoxazole compounds measurements well support the weight loss and electrochemical for protection of mild steel in hydrochloric acid solution. finding. Electrochim. Acta 55, 1720–1724. Musa, A.Y., Jalgham, R.T.T., Mohamad, A.B., 2012. Molecular Acknowledgement dynamic and quantum chemical calculations for phthalazine derivatives as corrosion inhibitors of mild steel in 1 M HCl. Corros. Sci. 56, 176–183. Chandra Bhan Verma gratefully acknowledges Ministry of Norr, E.A., 2008. Comparative study on the corrosion inhibition of Human Resource Development (MHRD), New Delhi (India) mild steel by aqueous extract of Fenugreek seeds and leaves in for providing financial assistance and facilitation for present acidic solutions. J. Eng. Appl. Sci. 3, 23–30. study. Ozkir, D., Kayakirilmaz, K., Bayol, E., Gurten, A.A., Kandemirli, F., 2012. The inhibition effects of Azure A on mild steel in 1 M HCl. A References complete study: adsorption, temperature, duration and quantum chemical aspects. Corros. Sci. 56, 143–152. Anejjar, A., Salghi, R., Zarrouk, A., Benali, O., Zarrok, H., Popova, A., Christov, M., 2006. Evaluation of impedance Hammouti, B., Ebenso, E.E., 2014. Inhibition of carbon steel measurements on mild steel corrosion in acid media in the presence corrosion in 1 M HCl medium by potassium thiocyanate. J. Assoc. of heterocyclic compounds. Corros. Sci. 48, 3208–3221. Arab Univ. Basic Appl. Sci. 15, 21–27. Quartarone, G., Battilana, M., Bonaldo, L., Tortato, T., 2008. Ansari, K.R., Quraishi, M.A., 2015. Effect of three component Investigation of the inhibition effect of indole-3-carboxylic acid (aniline–formaldehyde and piperazine) polymer on mild steel on the copper corrosion in 0.5 M H2SO4. Corros. Sci. 50, 3467– corrosion in hydrochloric acid medium. J. Assoc. Arab Univ. 3474. Basic Appl. Sci. 18, 12–18. Solmaz, R., 2014. Investigation of corrosion inhibition mechanism and Bahrami, M.J., Hosseini, S.M.A., Pilvar, P., 2010. Experimental and stability of vitamin B1 on mild steel in 0.5 M HCl solution. Corros. theoretical investigation of organic compounds as inhibitors for Sci. 81, 75–84. mild steel corrosion in sulfuric acid medium. Corros. Sci. 52, 2793– Sonar, S.S., Sadaphal, S.A., Kategaonkar, A.H., Pokalwar, R.U., 2803. Shingate, B.B., Shingare, M.S., 2009. Alum catalyzed simple and Bammou, L., Belkhaouda, M., Salghi, R., Benali, O., Zarrouk, A., efficient synthesis of bis(indolyl)methanes by ultrasound approach. Zarrok, H., Hammouti, B., 2014. Corrosion inhibition of steel in Bull. Korean Chem. Soc. 30, 825–828. sulfuric acidic solution by the Chenopodium Ambrosioides Extracts. Surasani, R., Kalita, D., Chandrasekhar, K.B., 2013. Indion Ina 225H J. Assoc. Arab Univ. Basic Appl. Sci. 16, 83–90. resin as a novel, selective, recyclable, eco-benign heterogeneous Barmatov, E., Hughes, T., Nagl, M., 2015. Efficiency of film-forming catalyst for the synthesis of bis(indolyl) methanes. Green Chem. corrosion inhibitors in strong hydrochloric acid under laminar and Lett. Rev. 6, 113–122. turbulent flow conditions. Corros. Sci. 92, 85–94. Verma, C.B., Quraishi, M.A., Singh, A., 2014a. 2-Aminobenzene-1,3- Daoud, D., Douadi, T., Hamani, H., Chafaa, S., Al-Noaimi, M., dicarbonitriles as green corrosion inhibitor for mild steel in 1 M 2015. Corrosion inhibition of mild steel by two new S- HCl: electrochemical, thermodynamic, surface and quantum chem- heterocyclic compounds in 1 M HCl: experimental and compu- ical investigation. J. Taiwan Inst. Chem. Eng. 2014, 1–11. http:// tational study. doi: http://dx.doi.org/10.1016/j.corsci.2015.01.025. dx.doi.org/10.1016/j.jtice.2014.11.029. Deng, S., Li, X., Fu, H., 2011. Acid violet 6B as a novel corrosion Verma, C.B., Reddy, M.J., Quraishi, M.A., 2014b. Microwave assisted inhibitor for cold rolled steel in hydrochloric acid solution. Corros. eco-friendly synthesis of chalcones using 2,4-dihydroxyacetophe- Sci. 53, 760–778. none and aldehydes as corrosion inhibitors for mild steel in 1 M Eddy, NO., Momoh-Yahaya, H., Oguzie, EE., 2014. Theoretical and HCl. Anal. Bioanal. Electrochem. 6, 321–340. experimental studies on the corrosion inhibition potentials of some Yadav, M., Behera, D., Kumar, S., Sinha, R.R., 2013. Experimental purines for aluminum in 0.1 M HCl. J. Adv. Res. doi: http://dx.doi. and quantum chemical studies on the corrosion inhibition perfor- org/10.1016/j.jare.2014.01.004. mance of benzimidazole derivatives for mild steel in HCl. Ind. Eng. Faustin, M., Maciuk, A., Salvin, P., Roos, C., Lebrini, M., 2015. Chem. Res. 52, 6318–6328. 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ORIGINAL ARTICLE Spectrophotometric method for pregabalin determination: An experimental design approach for method development

Dipak D. Patil a,*, Mukesh S. Patil a, Yogita B. Wani b a Department of Quality Assurance, H.R. Patel Institute of Pharmaceutical Education and Research, Karwand Naka, Shirpur, Dhule (Maharashtra State) 425405, India b Department of Pharmaceutical Chemistry, R.C. Patel Institute of Pharmaceutical Education and Research, Karwand Naka, Shirpur, Dhule (Maharashtra State) 425405, India

Received 12 September 2014; revised 6 January 2015; accepted 11 March 2015 Available online 15 June 2015

KEYWORDS Abstract A simple, sensitive and reproducible spectrophotometric method is developed for deter- Pregabalin; mining the pregabalin (PGB) content in bulk and in capsule dosage form using an experimental p-Dimethylamino- design approach. The proposed method is based on the condensation reaction of PGB (primary benzaldehyde; amine) with p-dimethylaminobenzaldehyde (PDAB) in an acidic medium to form a PGB–PDAB Experimental design complex. The PGB–PDAB complex shows maximum absorption at 395.80 nm. The proposed approach; method is validated according to the ICH Q2 (R1) guidelines for validation of analytical methods. OVAT The percentage purity of PGB in capsule dosage form as determined using the proposed method is 100.05 ± 1.48 whereas the corresponding value by the official method (Indian Pharmacopoeia, 2010) is 100.46 ± 0.41. The t-value and F-value are calculated for statistical comparison and are found to be 0.60 and 0.08, respectively. The proposed method may recommend for routine quality control analysis of PGB in its pharmaceutical dosage form. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction pharmacologically related to gabapentin (Indian Pharmacopoeia, 2010). PGB is officially approved in Indian Pregabalin (PGB), [S-[+]-3-isobutyl GABA or (S)-3-(aminome Pharmacopoeia (2010) and the use of a liquid chromatographic thyl)-5-methylhexanoic acid, trade name Lyrica] is an anticon- (LC) method for determining the PGB content in bulk and cap- vulsant and analgesic medication that is structurally and sule dosage form has been described. Various LC methods have been reported in the literature (Vermeij and Edelbroek, 2004; Dousˇ a et al., 2010; Kannapan et al., 2010; Karavadi and * Corresponding author. Tel.: +91 9960878673; fax: +91 2563 Challa, 2014). A number of UV spectrophotometric methods 257599. E-mail addresses: [email protected] (D.D. Patil), have also been developed for determining PGB content using [email protected] (M.S. Patil), [email protected] different derivatizing reagents. The reagents include potassium (Y.B. Wani). iodate and potassium iodide (Gujral et al., 2009), fluo- Peer review under responsibility of University of Bahrain. rescamine, 2,4-diclorofluorobenzene and 2,3,5,6-tetrachloro-1, http://dx.doi.org/10.1016/j.jaubas.2015.03.002 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 32 D.D. Patil et al.

4-benzoquinone (Shaalan, 2010), ninhydrin (Bali and Gaur, capsule dosage form of PGB (PREGEB 75) was purchased 2011), 1,2-naphthoquinone-4-sulfonate and 2,4-dinitrofluoro from a local medical shop. A Shimadzu UV–visible spec- benzene (Walash et al., 2011), ascorbic acid and salicylaldehyde trophotometer (model 1700) with a pair of matched 1 cm (Hammad and Abdallah, 2012) and bromocresol green (Merin quartz cells and UV-Probe software was used for absorbance et al., 2013). In the current literature Ismail et al. (2016) and measurements. Kumble and Narayana (2016) also developed spectrophoto- metric methods for the determination of bifonazole and feban- 2.1. Reagents and solutions tel, respectively. In methods mentioned in the foregoing, the experimental variables, such as the reagent concentration, 2.1.1. PDAB solution (1.5%, w/v) reagent volume, heating temperature, heating time and pH of An accurately weighed quantity of about 150 mg of PDAB was the reaction mixture were optimized using the one variable at dissolved in 10 mL of methanol. The solution was freshly pre- a time method (OVAT). As the name suggests, one experimental pared and protected from light. variable is optimized at a time by varying its concentration or value within a given range, keeping the values of the remaining 2.1.2. PDAB solution (1M) parameters constant. So, a few parameters are not optimized. Because of this reason the OVAT method gives misleading An accurately weighed quantity of about 1.63 g of PDAB was results and is inefficient for optimizing experimental variables dissolved in 10 mL of methanol. (Lewis et al., 1999). Hence, the use of this method for optimiz- ing experimental variables must be avoided. 2.1.3. Mobile phase Another disadvantage of the OVAT method is that the A ternary mobile phase consisting of 0.022 M potassium dihy- number of experiments necessary is large, as a result of which drogen phosphate (pH 6), methanol and acetonitrile in the the time required and expenses involved are high, with a high proportion of 92:5:3, v/v/v was used for HPLC analysis. consumption of reagents and materials in the experiments. Mobile phase was used as diluent for HPLC analysis. A systematic and statistical system i.e. an experimental design approach is needed to overcome the limitations of the 2.1.4. Standard solution of PGB (1000 lg/mL) OVAT method and optimize experimental variables so as to An accurately weighed quantity of about 10.0 mg PGB was obtain significant and precise analytical results. Therefore, in dissolved in 10 mL of methanol. the present study we used the experimental design approach for screening and optimizing the experimental variables 2.1.5. Standard solution of PGB (1M) involved in the reaction between PGB and PDAB. PDAB is An accurately weighed quantity of about 1.58 g PGB was dis- used to derivate PGB because it lacks chromophore. solved in 10 mL of methanol. Previously we had determined the content of cefixime trihy- drate using ninhydrin as a derivatizing agent (Wani and 2.1.6. Standard solution of PGB for HPLC analysis (4000 Patil, 2013). The method was based on the reaction of the lg/mL) amino group of cefixime with ninhydrin in an alkaline medium An accurately weighed quantity of about 40.0 mg of PGB was to form a yellow-colored derivative (kmax 436 nm). The exper- imental design approach was utilized to screen and optimize dissolved in 10 mL of the mobile phase. experimental variables involved in the development of UV– Visible spectrophotometric method. 2.1.7. Sample solution of PGB for HPLC analysis (4000 In the current work, we planned to determine the PGB con- lg/mL) tent in bulk and in a pharmaceutical dosage form using PDAB The contents of 20 capsules were removed and weighed. An as the derivatizing reagent. An amino compound that lacks amount of this powder equivalent to 40.0 mg of PGB was chromophore can be assayed spectrophotometrically using a transferred to a 10 mL volumetric flask (flask). The mobile suitable carbonyl reagent. One of the most frequently used phase was added, and the contents of the flask were sonicated reagent PDAB, known as Ehrlich’s reagent (Beckett and for 10 min. The volume in the flask was made up to the mark Stenlake, 1997). Therefore, we selected PDAB as the derivatiz- with the mobile phase. The solution was filtered through a ing reagent for PGB in the present work. The aim of the present 0.45 lm filter paper. work was to utilize an experimental design approach for devel- oping a UV–visible spectrophotometric method for determining 2.2. Optimization of reaction variables using experimental the PGB content using PDAB as the derivatizing reagent. The design approach screening of the experimental variables and their optimization was carried out through a two level fractional factorial design A two level fractional factorial design was used to screen the and a response surface methodology, respectively (Minitab effect of the PDAB reagent concentration (% w/v) (X1 (A)), Release, 1997). volume of PDAB reagent (ml) (X2 (B)), volume of concen- trated HCl (ml) (X3 (C)), heating temperature (C) (X4 (D)) 2. Materials and methods and heating time (minutes) (X5 (E)) on absorbance (R) of the PGB–PDAB complex. On the basis of preliminary experiments A PGB sample was obtained from Wockhardt Pharmaceutical performed using the conventional method, the following Ltd., Aurangabad, India. PDAB and analytical grade concen- ranges of values were used in the design-X1: 1–2% w/v; X2: trated hydrochloric acid (HCl) were purchased from Loba 1.5–3.5 mL; X3: 0.05–0.15 mL; X4: 50–70 C and X5:5– Chemie, Mumbai, India. A commercial pharmaceutical 15 min (Table 1). The design matrix of the two-level fractional Spectrophotometric method for pregabalin determination 33

2.2.3. Linearity study Table 1 Experimental factors and response variable for two level full factorial design Aliquots of the standard solution of PGB (about 0.05–0.6 mL) were transferred into a series of 10 mL flasks. To each flask, Experimental factors Code Levels P-value 2.5 mL of 1.5% w/v PDAB solution and 0.1 mL of concen- Low High trated HCl were added. The solution was mixed thoroughly,

PDAB reagent concentration X1 1 2 0.084 heated on a water bath at 60 C for 10 min. The reaction mix- (% w/v) ture was cooled in an ice bath and the final volume was made

Volume of PDAB reagent (ml) X2 1.5 3.5 0.973 up to the mark with methanol. The solution was mixed, and Volume of Conc. HCl (ml) X3 0.05 0.15 0.741 the absorbance was measured at 395.80 nm against a blank. Heating temperature (C) X4 50 70 0.695 Heating time (min) X5 5 15 0.026 2.2.4. Assay procedure Response R Formation of PGB–PDAB Maximum The content of 20 capsules was weighed. An amount of the complex (absorbance) powder equivalent to 25 mg of PGB was transferred to a 25 mL flask. Methanol was added and the contents of the flask were sonicated for 10 min so as to dissolve them. The volume was made up to the mark in the flask with methanol. The solu- factorial design (1/2 fraction, V resolution, 2(5-1) and 16 runs) is tion was filtered through Whatman filter paper. Aliquot of shown in Table 2. All the experimental runs were performed in about 0.1 mL of the solution was transferred to five 10 mL triplicate and the average absorbance is provided in Table 2. flasks. About 2.5 mL of PDAB solution (1.5% w/v) and 0.1 mL of concentrated HCl was added to each flask. The solu- 2.2.1. Optimization using response surface methodology tion was mixed thoroughly and heated in a water bath at 60 C A three-level central composite design (Table 3) was used to for 10 min and cooled in an ice bath. The volume was then optimize X1 and X5. The following experimental conditions made up to the mark with methanol. The solution was mixed were maintained constant-X2: 2.5 mL; X3: 0.1 mL; X4:60C and absorbance at 395.80 nm measured against a blank. The and diluting solvent:methanol. The experiment was performed PGB content in the pharmaceutical dosage form was calcu- in triplicate and the average absorbance is provided in Table 3. lated using the following formulae: The optimized experimental variables, as determined using the ðAU WStd WAvgÞ fractional factorial and three level central composite designs XEst ¼ ðAS WSamÞ were used in the subsequent experiments.

XEst 2.2.2. Optimization of reaction variables by conventional method %Labeled claim ¼ 100 XLC (OVAT) where X = content of PGB present in powder per capsule; Different experimental parameters, such as X , X , X , X and Est 1 2 3 4 A = absorbance of the sample solution; A = absorbance X , were optimized using the conventional method to obtain U S 5 of the standard solution; W = weight of the standard the maximum reaction product or absorbance (R). The Std (mg); W = weight of the sample (mg); W = average resulting solutions were scanned in 200–800 nm range. The Sam Avg weight of powder in capsule (mg); and X = labeled content maximum absorbance of the complex lies in this range, at LC of PGB per capsule (mg). 395.80 nm. 2.2.5. Assay procedure of official method A LC method is recommended in the Indian Pharmacopoeia, 2010 for determining the PGB content in a capsule dosage Table 2 Design matrix of fractional factorial design and form. Analysis of the PGB was carried out on a Luna C18 col- response value umn (250 mm · 4.6 mm, 5 lm) using a ternary mobile phase consisting of 0.022 M potassium dihydrogen phosphate (pH Run X1 X2 X3 X4 X5 R (n =3) 6), methanol and acetonitrile (92:5:3, v/v/v) at a flow rate of 1 2 3.5 0.15 70 15 0.651 1 mL/min. The standard and sample solutions were injected 2 2 1.5 0.05 70 15 0.630 for HPLC analysis. Detection of PGB was carried out at 3 2 1.5 0.05 50 5 0.715 205 nm. The chromatogram was recorded for PGB. This 4 1 1.5 0.15 50 5 0.632 showed a retention time of 13.42 min. Calculations were made 5 2 3.5 0.05 50 15 0.571 from the peak areas of the standard and sample 6 2 1.5 0.15 50 15 0.622 7 1 1.5 0.15 70 15 0.588 chromatograms. 8 1 3.5 0.15 50 15 0.561 9 1 1.5 0.05 70 5 0.695 2.2.6. Accuracy 10 2 3.5 0.05 70 5 0.721 The accuracy of the proposed method was determined through 11 1 3.5 0.15 70 5 0.531 a recovery study. A known amount of pure PGB was spiked to 12 1 1.5 0.05 50 15 0.493 pre-analyzed capsule formulation. Analysis of PGB was 13 2 3.5 0.15 50 5 0.741 carried out at concentrations of 80%, 100% and 120% and 14 1 3.5 0.05 50 5 0.681 15 1 3.5 0.05 70 15 0.500 working solution concentration of about 18 lg/mL, 16 2 1.5 0.15 70 5 0.571 20 lg/mL and 22 lg/mL, respectively. Determination of the PGB content was carried out using the formula mentioned in 34 D.D. Patil et al.

Table 3 Design matrix for optimization of method parameters

StdOrder RunOrder PtType Blocks X1 X5 R(n =3) 11 1 0 1 0 0 0.870 13 2 0 1 0 0 0.875 12 3 0 1 0 0 0.891 4 4 1 1 1 1 0.886 10 5 0 1 0 0 0.893 661 1 1.414214 0 0.785 7711 0 1.41421 0.675 18 111 1 0.676 9 9 0 1 0 0 0.642 210111 1 0.640 811 1 1 0 1.414214 0.588 312111 1 0.565 513 11 1.41421 0 0.536

the section 2.2.4 entitled ‘‘Assay procedure’’. The percentage present work. The experimental design approach was used to recovery of the proposed method was calculated using the fol- screen and optimize the experimental variables involved in lowing formula: the proposed method. The reaction took place between the E carbonyl functional group of PDAB and the free amino group %Recovery ¼ 100 of PGB yields Schiff’s base. The reaction proceeds via an T þ P attack of the –NH2 group on carbonyl carbon of the where E = total amount of PGB estimated (mg); T = amount O‚CHA to form AN‚CA with the elimination of a water of PGB from pre-analyzed powder of the capsule (mg); and molecule (Fig. 1)(Beckett and Stenlake, 1997). The PGB– P = amount of pure PGB added (mg). PDAB reaction product showed the maximum absorption at 395.80 nm against the blank (Fig. 1). 2.2.7. Precision The inter-day precision and intra-day precision of the method 3.1. Optimization of reaction variables using experimental were determined. A repeatability study (intra-day precision) design approach was performed by analyzing a PGB solution (10 lg/mL) repeatedly within a day. An inter-day precision study was per- A two level fractional factorial design or Plackett–Burman formed by analyzing a PGB solution (10 lg/mL) repeatedly on design (1/2 fraction, V resolution, 2(5-1) and 16 runs) was used different days. to evaluate the main effects of five independent factors on the selected response, R. From the Pareto chart of the effects, it 2.2.8. Limit of detection and limit of quantitation was observed that factors X1 and X5 had direct effect on the The limit of detection (LOD) and limit of quantitation (LOQ) were calculated using the following formulae: LOD ¼ð3:3 rÞ=S

LOQ ¼ð10 rÞ=S where r = standard deviation of the response; and S = slope of calibration curve.

2.2.9. Stoichiometry study A specific volume (about 0.2 mL) of PGB solution (1 M) was transferred to a series of 10 mL flasks. PDAB solution (1 M) was transferred serially (0.2 mL to 3.4 mL in 0.2 mL steps) into the flasks. Concentrated HCl (0.1 mL) was added to each flask. The reaction mixture was thoroughly mixed and heated on a water bath at 60 C for 10 min. The reaction mixture was cooled in an ice bath and the final volume was made up to the mark with methanol. The absorbance of colored solution was measured at 395.80 nm against the reagent blank.

3. Result and discussion

A spectrophotometric method was optimized for determining PGB content using PDAB as a derivatizing reagent in the Figure 1 Reaction and UV spectrum of PGB with PDAB. Spectrophotometric method for pregabalin determination 35 absorbance, whereas factor X2, X3 and X4 also had direct effects on the absorbance but not significant (Fig. 2). From a normal and half normal plot of the effects and the P value, fac- tors X1 and X5 were found to be significant (P value < 0.1) and factor X2, X3 and X4 were insignificant (P value > 0.1) (Table 1).

3.1.1. Optimization using response surface methodology Using multivariate regression analysis, a fitted full quadratic model was obtained for the average response, R, which is given by the following equation:

2 2 R ¼ b0 þ b1X1 þ b5X5 þ b11X1 þ b55X5 þ b15X1X5 ð1Þ where R = selected response; b0 = arithmetic mean response; b1 and b5 = regression coefficients of the factors X1 and X5, respectively. A central composite design was used to optimize the reac- tion variables (X1 and X5). From the response surface plot and contour plot (Fig. 3), it was observed that the absorbance increases from the lower left corner to the upper right corner of Figure 3 Response surface plot and contour plot of response R. the plot. In other words, the absorbance increases as the con- centration of PDAB and heating time increases simultane- ously. This plot suggests that the maximum absorbance was Table 4 Analysis of variance for response obtained with the concentration range between 1.5 and 2% Source DF Seq SS Adj SS Adj MS FP w/v of the PDAB and a heating time of 10–15 min (Fig. 3) (Table 3). Above a concentration of 1.5% w/v PDAB reagent Regression 5 0.199645 0.199645 0.039929 40.69 0.000 Linear 2 0.046618 0.046618 0.023309 23.75 0.001 and more than 10 min heating, the absorbance was constant. X 1 0.044698 0.044698 0.044698 45.55 0.000 Therefore, a 1.5% w/v concentration of PDAB reagent and 1 X5 1 0.001920 0.001920 0.001920 1.96 0.205 a heating time of 10 min are the optimized conditions. Square 2 0.123956 0.123956 0.061978 63.16 0.000

Tables 4 and 5 show the values of the regression coefficients X1 * X1 1 0.044423 0.060394 0.060394 61.55 0.000 and their associated P-values. From Tables 4 and 5, it may X1 * X5 1 0.079534 0.079534 0.079534 81.05 0.000 be observed that the concentration of PDAB affected the Interaction 1 0.029070 0.029070 0.029070 29.63 0.001 response, R, significantly (P value < 0.1). X1 * X5 1 0.029070 0.029070 0.029070 29.63 0.001 The plot of the main effects indicates that the PDAB Residual 7 0.006869 0.006869 0.000981 reagent concentration and heating time have similar effects error on the response. For both the factors, the response increases Lack-of- 3 0.003789 0.003789 0.001263 1.64 00315 fit as you move from a low level to a medium level, and the Pure error 4 0.003079 0.003079 0.000770 response decreases from a medium level to a high level of the Total 12 0.206513

Table 5 Estimated regression coefficients for response Term Coef SECoef TP Constant 0.87960 0.01401 62.789 0.000

X1 0.07475 0.01107 6.749 0.000 X5 0.01549 0.01107 1.399 0.205 X1 * X1 0.09318 0.01188 7.845 0.000 X5 * X5 0.10692 0.01188 9.003 0.000 X1 * X5 0.08525 0.01566 5.443 0.001

factor. However, the interaction plot shows that the increase in the response is greater when the PDAB reagent concentra- tion is in the range between 1.5% w/v and 2.0% w/v than when the PDAB reagent concentration is in the ranges between 0.79% w/v to 0.1% w/v and 2.0% w/v to 2.2% w/v. The developed model was validated. The experimental results and the predicted values obtained using the polynomial model equation showed that the predicted value matches rea- sonably with the R-Sq value of 96.67% and R-Sq (adj) of Figure 2 Pareto chart of the effects. 94.30% of the selected response, R. 36 D.D. Patil et al.

The distribution of the residuals for the response approxi- mately followed the fitted normal distribution, whereas the residuals of the response were randomly scattered in the residual plots. In the present work, the experimental variables were optimized using the conventional method (OVAT) also. The optimum conditions required for the maximum color development were found to be 2.5 mL of 1.5% w/v PDAB solution, 0.1 mL of concentrated HCl and 10 min of heating at 60 C.

3.1.2. Assay and statistical comparison The results of the analysis of the marketed formulation using the proposed and official methods are presented in Table 6. The percentage purity of the commercial formulation PREGEB 75 capsule as determined using the proposed method was 100.05 ± 1.48 and that determined using the official method was 100.46 ± 0.41. The two results were com- pared by calculating the t-value and F-value, which were found to be 0.60 and 0.08, respectively. From these results, it is Figure 4 Stoichiometry between PGB and PDAB. concluded that there is no significant difference between the proposed and official methods, indicating that the proposed method is as accurate and precise as the official method. 3.2. Method validation

3.1.3. Stoichiometry study A calibration graph was constructed with PGB concentration In this study the weight relationship between PGB, PDAB and on the X-axis and absorbance on the Y-axis using all the the PGB–PDAB complex was determined. The ratio of PGB optimized conditions described in the foregoing. A linear rela- to PDAB in the complex was determined using the molar ratio tionship was found in the concentration range from method (Miller and Miller, 1993). A graph with a number of 5–60 lg/mL (r2 = 0.9960) (Table 6). The accuracy study, indi- moles of PDAB per mole of PGB on the X-axis and absor- cated that the average percentage recovery for PREGEB 75 bance on the Y-axis shows that a molar ratio of 4:1 capsules was 101.03 ± 0.94 (Table 6). The inter-day and intra- (PDAB:PGB) is sufficient for maximum color development day precision values were 1.07 and 0.66 lg/mL, respectively. (Fig. 4). The % RSD value obtained (<2) indicates that the proposed method is precise. The LOD and LOQ were found to be 0.025 lg/mL and 0.076 lg/mL, respectively. The solution used in the precision study was stored for 24 h at room temperature. During the storage period, the absorbance of this solution was Table 6 Optical characteristics, assay result and statistical measured at different time intervals (0 min, 15 min, 30 min, data of the validation parameters 1 h, 2 h, 3 h, 4 h, 5 h and 24 h). The color intensity of the solu- Parameter Observation tion was stable up to 6 h. Medium Acidic Color Yellow 4. Conclusion

kmax (nm) 395.80 Beer–Lambert’s law limit (lg/ml) 5–60 In the present work optimization of a spectrophotometric Slope 1.252 method for determining PGB content was optimized using Intercept 0.033 Regression coefficient 0.9960 experimental design methodology and the conventional method (OVAT). In the factorial design, out of five factors Assay results (n =5) (X1, X2, X3, X4 and X5), two factors (X1 and X5) were found Proposed method 100.05 ± 1.48 to be significant. These significant factors were optimized using Reported method 100.46 ± 0.41 response surface methodology. The results of optimization of t-valuea 0.60 F-valueb 0.08 reaction variables by both the methods were found to be identical ones. However, the experimental design approach Accuracy (n =3) was systematic, less time consuming and more cost effective Level 80% 101.13 ± 0.15 compared with the conventional method. Therefore, the pro- 100% 101.89 ± 0.90 posed method using an experimental design approach is better 120% 100.01 ± 1.76 Mean 101.03 ± 0.94 for determining the PGB content and can be used for routine quality control analysis of PGB formulations. Precision (%RSD, n =5) Intraday 0.66 Acknowledgments Interday 1.07 a t0.05%, 4 = 2.776. We are thankful to Wockhardt Pharmaceutical Ltd., b F0.05%, 4 = 6.39. Aurangabad (India) for providing gift sample of PGB for Spectrophotometric method for pregabalin determination 37 our research work. We would also like to extend our thanks to Karavadi, T.M., Challa, B., 2014. Bioanalytical method development Dr. S.B. Bari, Principal, H.R. Patel Institute of Pharmaceutical and validation of pregabalin in rat plasma by solid phase extraction Education and Research, Shirpur for providing research with HPLC-MS/MS: application to a pharmacokinetic study. J. facilities to carry out this work. Liquid Chromatogr. Relat. Technol. 37 (1), 130–144. Kumble, D., Narayana, B., 2016. A novel spectrophotometric approach for the determination of febantel in pure and dosage References forms. J. Assoc. Arab Univ. Basic Appl. Sci. 19, 23–28. Lewis, G.A., Mathieu, D., Phan-Tan-Luu, R., 1999. Pharmaceutical Bali, A., Gaur, P., 2011. A novel method for spectrophotometric Experimental Design. Marcel Dekker Inc., New York. determination of pregabalin in pure form and in capsules. Chem. Merin, K.O., Cicy, S.E., Sheeja, V., 2013. Validated spectrophoto- Cent. J. 5 (1), 1–7. metric method for the determination of pregabalin in pharmaceu- Beckett, A.H., Stenlake, J.B., 1997. Practical Pharmaceutical ticals based on charge transfer reaction. Int. J. Pharm. Res. Dev. 5 Chemistry. CBS Publishers and Distributors Pvt. Ltd., New Delhi. (2), 42–48. Dousˇ a, M., Gibala, P., Lemr, K., 2010. Liquid chromatographic Miller, J.C., Miller, J.N., 1993. Statistics for Analytical Chemistry, separation of pregabalin and its possible impurities with fluores- third ed. Ellis Horwood, New York. cence detection after postcolumn derivatization with o-phtaldialde- Minitab Release, 1997. User guide: Data analysis and quality tools, hyde. J. Pharm. Biomed. Anal. 53 (3), 717–722. Minitab Inc. Gujral, R.S., Haque, S.M., Shanker, P., 2009. A sensitive spectropho- Shaalan, R.A.A., 2010. Spectrofluorimetric and spectrophotometric tometric method for the determination of pregabalin in bulk, determination of pregabalin in capsules and urine samples. Int. J. pharmaceutical formulations and in human urine samples. Int. J. Biomed. Sci. 6 (3), 260–267. Biomed. Sci. 5 (4), 421–427. Vermeij, T.A.C., Edelbroek, P.M., 2004. Simultaneous high-perfor- Hammad, S.F., Abdallah, O.M., 2012. Optimized and validated mance liquid chromatographic analysis of pregabalin, gabapentin spectrophotometric methods for the determination of pregabalin and vigabatrin in human serum by precolumn derivatization with in pharmaceutical formulation using ascorbic acid and salicylalde- o-phtaldialdehyde and fluorescence detection. J. Chromatogr. B hyde. J. Am. Sci. 8 (12), 118–124. 810 (2), 297–303. Indian Pharmacopoeia, 2010. Ghaziabad, Indian Pharmacopoeial Walash, M.I., Belal, F.F., El-Enany, N.M., El-Maghrabey, M.H., Commission. 2011. Utility of certain nucleophilic aromatic substitution reactions Ismail, N.B.S., Narayana, B., Kumble, D., 2016. Validated spec- for the assay of pregabalin in capsules. Chem. Central J. 5 (1), 1–10. trophotometric methods for the determination of bifonazole in Wani, Y.B., Patil, D.D., 2013. An experimental design approach for pharmaceuticals by charge transfer complexation. J. Assoc. Arab optimization of spectrophotometric method for estimation of Univ. Basic Appl. Sci. 19, 8–14. cefixime trihydrate using ninhydrin as derivatizing reagent in bulk Kannapan, N., Nayak, S.P., Venkatachalam, T., Prabhakaran, V., and pharmaceutical formulation. J. Saudi Chem. Soc. http:// 2010. Analytical RP-HPLC method for development and valida- dx.doi.org/10.1016/j.jscs.2013.11.001. tion of pregabalin and methylcobalamine in combined capsule formulation. Appl. Chem. Res. 13, 85–89. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016) 21,38–44

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ORIGINAL ARTICLE Characterization and antibacterial activity of nanocrystalline Mn doped Fe2O3 thin films grown by successive ionic layer adsorption and reaction method

M.R. Belkhedkar a,b, A.U. Ubale a,*, Y.S. Sakhare a, Naushad Zubair c, M. Musaddique c a Nanostructured Thin Film Materials Laboratory, Department of Physics, Govt. Vidarbha Institute of Science and Humanities, VMV Road, Amravati 444604, Maharashtra, India b Department of Physics, Shri Shivaji College, Akola 444003, Maharashtra, India c Department of Microbiology, Shri Shivaji College, Akola 444003, Maharashtra, India

Received 25 November 2014; revised 24 February 2015; accepted 11 March 2015 Available online 8 June 2015

KEYWORDS Abstract Successive ionic layer adsorption and reaction (SILAR) method have been successfully Thin film; employed to grow nanocrystalline Mn doped a-Fe2O3 thin films onto glass substrates. The struc- Nanostructures; tural analysis revealed that, the films are nanocrystalline in nature with rhombohedral structure. Biomaterials; The optical studies showed that a-Fe2O3 thin film exhibits 3.02 eV band gap energy and it decreases Optical properties; to 2.95 eV as the Mn doping percentage in it was increased from 0 to 8 wt.%. The SILAR grown

Antibacterial activity a-Fe2O3 film exhibits antibacterial character against Staphylococcus aureus bacteria and it increases remarkably with Mn doping. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Mulenko et al., 2012; Rahman and Joo, 2013; Cavas et al., 2013; Akhavan, 2010; Hsien et al., 2013; Kitaura et al., 2008; Kulal et al., 2011; Ji et al., 2011). In recent years, many Since the past decades, a-Fe2O3 has gained extensive scientific importance in materials science because of its important role in researchers have studied the role of doping in a-Fe2O3 to various applications namely gas sensor, supercapacitor, dye improve its applicability for electrochemical sensors, solid sensitized solar cell, photocatalyst, lithium ion battery and in oxide fuel cell and photo splitting of water etc. (Suresh et al., microbial fuel cells (Lee et al., 2001; Fan et al., 2011; 2012; Geng et al., 2012; Shwarsctein et al., 2008). In addition, Shinde et al. (2011) have reported structural, morphological, luminescent and electronic properties of spray deposited Al * Corresponding author. Tel.: +91 721 2531706; fax: +91 721 doped a-Fe2O3 thin films. Khan and Zhou (1993) have 2531705. reported physical properties of iodine doped a-Fe2O3 thin E-mail address: [email protected] (A.U. Ubale). films grown by spray pyrolysis. Sanchez et al. (1988) have Peer review under responsibility of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2015.03.001 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Nanocrystalline Mn doped a-Fe2O3 thin film growth by SILAR method 39 reported photoelectrochemical properties of niobium doped 160 a-Fe2O3 thin films grown by chemical vapor deposition Fe2O3 method. Sensing properties of LPD synthesized Pd and Ca Mn3O4 doped a-Fe2O3 thin films were reported by Neri et al. (2007). 120 As per literature to grow doped and undoped a-Fe2O3 thin films, various chemical deposition methods have been utilized viz. spin coating deposition (Souza et al., 2009), liquid-phase 80 method (Neri et al., 2001), electrodeposition (Kumar et al., 2011), spray-pyrolysis (Kumari et al., 2010) and DC sputtering (Kiran et al., 2006). The growth of nanostructured thin films 40 by a simple and economic deposition technique has been Thickness of thin films (nm) playing an important role in the field of nanoscience and nanotechnology in order to reduce the device fabrication cost. 0 In this paper, we have outlined the simple and economic 0306090 successive ionic layer adsorption and reaction method to grow Number of SILAR cycles nanocrystalline Mn doped a-Fe2O3 thin films. This is an excellent method to grow a nanocrystalline thin film by Figure 1 Plot of a-Fe2O3 and Mn3O4 film thickness with number immersing the glass substrate into separately placed cationic of SILAR cycles [Cationic Source: 0.06 M FeCl3 (for Fe2O3) and and anionic precursors alternately. In between each immer- 0.3 M MnCl2 (for Mn3O4)]. sion, the substrate is rinsed in deionized water to remove the loosely bound species to get an adherent thin film. As cationic 8 wt.% Mn doped Fe2O3 and anionic precursors are separately placed, it is very easy to 6 wt.% Mn doped Fe2O3 control the growth process by adjusting the number of deposi- 4 wt.% Mn doped Fe2O3 2 wt.% Mn doped Fe2O3 tion cycles. The structural, morphological and optical proper- undoped Fe2O3 *( 0 1 2) ties of nanocrystalline Mn doped a-Fe2O3 thin films grown by the SILAR method are discussed in the paper. Also, the results *( 1 0 4) #(1 2 0) of antibacterial activity of the Mn doped a-Fe2O3 films against #(0 4 5) *( 1 6) Staphylococcus aureus bacteria are reported. *(0 1 8)

2. Experimental Intensity (A.U.)

2.1. Preparation of Mn doped a-Fe2O3 thin films

To grow Mn doped a-Fe2O3 thin films 0.3 M MnCl2 of pH 1 and 0.06 M FeCl2 of pH <1 were used as cationic precursors 20 40 60 80 along with 0.005 M NaOH (pH 12) solution as an anionic 2θ (degree) precursor. The beakers containing precursor solutions were placed alternately in such a way that, every cationic and anionic Figure 2 GIXRD patterns of Mn doped a-Fe2O3 thin films. precursor was followed by a beaker containing deionized water. Several deposition trials were performed to optimize the vari- ous deposition parameters to grow Mn O and a-Fe O films 3 4 2 3 Table 1 Film thickness and crystallite size of a-Fe O with separately. For a-Fe O film, a well cleaned glass substrate 2 3 2 3 Mn doping percentage. was immersed in a cationic precursor for 20 s where Fe3þ ions were adsorbed on the substrate surface. The substrate was then wt.% of Mn Film thickness Average crystallite size from in a-Fe O (nm) rinsed in deionized water for 20 s to remove loosely bound Fe3þ 2 3 GIXRD (nm) FESEM (nm) ions. Finally it was immersed in NaOH solution for 20 s, where 0 105 21 28 3þ OH ions react with Fe ions to form a-Fe2O3 species. This 2 108 21 26 was again followed by rinsing in deionized water for 20 s to 4 107 20 23 remove loose material from the substrate surface. This com- 6 105 19 21 8 106 17 20 pletes one SILAR deposition cycle for a-Fe2O3. The immersion and rinsing time periods were experimentally optimized to get uniform and adherent thin films. The optimized deposition parameters for Mn3O4 films were already explained in our ear- that up to 50 SILAR cycles the growth rate was approximately lier report (Ubale et al., 2012). To achieve proper doping, sev- the same. Above 50 SILAR cycles, thickness of Mn3O4 film eral deposition trials were performed by varying the decreases as it peels off; however, the thickness of a-Fe2O3 film concentration of Fe precursor to match the growth rate of a- increases up to 60 SILAR cycles. Hence for the present work, to Fe2O3 film formation with Mn3O4. It was observed that, the grow Mn doped a-Fe2O3 thin films 50 SILAR cycles were growth rate of a-Fe2O3 and Mn3O4 is approximately the same considered. To achieve 0, 2, 4, 6 and 8 wt.% doping of Mn in for 0.06 and 0.3 M concentrations of FeCl3 and MnCl2 respec- a-Fe2O3, the number of SILAR cycles for (Fe2O3:Mn3O4) com- tively. Fig. 1 shows the variation of Mn3O4 and a-Fe2O3 film position were taken as (50:0), (49:1), (48:2), (47:3) and (46:4) thickness with a number of deposition cycles. It was observed respectively. These films were further annealed at 500 C for 40 M.R. Belkhedkar et al.

3 h to get a pure metal oxide phase of the deposited material film in cm2 and ‘q’ is the density of the deposited material. It and used for further characterization. was observed that, the thickness of Mn doped a-Fe2O3 thin films deposited by repeating 50 SILAR cycles is of the order 2.2. Characterization techniques of 105 nm. The crystal structure of the deposited film was identified by grazing incidence X-ray diffraction with Xpert In the present work, thickness of the film was measured by PRO PANalytical diffractometer. The surface morphological gravimetric weight difference method by using the relation, investigations were carried out by using field emission scanning electron microscope (Model: SUPRA 40) and atomic force m t ¼ ð1Þ microscope (Model: Nanonics Multiview 2000, Israel). The q A optical absorption studies were carried out in the wavelength where ‘m’ is the mass of the deposited material measured by range 350–750 nm using ELICO Double Beam SL 210 using a sensitive microbalance; ‘A’ is the area of the deposited UV–Vis spectrophotometer.

Figure 3 FESEM images of Mn doped a-Fe2O3 thin films; Mn doping percentage: (A) 0 wt.%, (B) 2 wt.%, (C) 4 wt.%, (D) 6 wt.% and (E) 8 wt.%. Nanocrystalline Mn doped a-Fe2O3 thin film growth by SILAR method 41

2.3. Antibacterial test The FESEM analysis shows compact and homogeneous distri- bution of nano grains of varying sizes from 28 to 20 nm A spread plate technique was employed to investigate antibac- depending upon doping percentage. The agglomeration of nano grains is observed at several places on the film surface. terial behavior of Mn doped a-Fe2O3 thin films against S. aur- eus. For it, the culture of S. aureus bacteria was prepared in It is seen that the number of agglomerated grains reduces with nutrient broth. The loopful culture of S. aureus organisms Mn doping. The grain sizes estimated from FESEM images are was further inoculated into 20 mL sterilized nutrient broth in good agreement with GIXRD results (Table 1). and incubated at 310 K for 24 h. Then, 20 lL cultures of S. The elemental analysis of the undoped and 6 wt.% Mn aureus was inoculated on undoped and Mn doped iron oxide doped a-Fe2O3 thin films were carried out using energy disper- glass substrates of area 1 cm2 with the help of an inoculating sive X-ray (EDX) analysis (Fig. 4). The elemental analysis was needle. These glass slides were then placed in previously steril- carried only for Fe, O and Mn elements. The additional peaks ized petri dishes and incubated at 310 K for 24 h. After incuba- observed in the spectra are due to the composition of the glass tion these slides were transferred to 3 mL of buffer peptone substrate. The small minor peak of Mn observed in the spectra solution in a test tube and ultrasonicated to detach the bacteria confirms its doping in a-Fe2O3. thoroughly from the substrate. From this, 20 lL washed buffer Fig. 5 shows 3D AFM images of Mn doped a-Fe2O3 thin peptone solution was then inoculated on nutrient agar plates films. The images were recorded in tapping mode by using by a spread plate technique and incubated at 310 K for 24 h an optical fiber tip, coated with gold and chromium (Au, Cr) to obtain viable bacteria. After successful incubation the viable metal with u = 20 nm at response frequency 52.38 kHz. The bacterial colonies were counted and antibacterial efficiency AFM analysis showed that the granular structure of a-Fe2O3 was calculated using the relation (Zhang et al., 2008), is uniformly distributed over the entire substrate surface. The morphological parameters such as rms roughness (Rq), aver- ðN NÞ r ¼ 0 100% ð2Þ age roughness (Ra), average height, maximum height and N0 grain orientation depending on doping percentage of Mn are listed in Table 2. The rms roughness of a-Fe O film decreases where, ‘r’ is the antibacterial efficiency, ‘N0’ is the number 2 3 of viable bacterial colonies observed in the petri dish from as doping concentration of Mn rises from 0 to 8 wt.%. The undeposited film and ‘N’ is the number of viable bacterial colo- rms roughness and average roughness becomes approximately equal above 6 wt.% doping of Mn which may be because of nies observed in the petri dish from Mn doped Fe2O3 thin film. uniform mixing of Mn3O4 in a-Fe2O3. 3. Results and discussion 3.3. Optical studies 3.1. Structural studies The optical absorption measurements for a-Fe2O3 thin films deposited onto glass substrates were carried out in the wavelength Fig. 2 shows typical GIXRD patterns of Mn doped iron oxide thin films. The pattern revealed that SILAR grown Fe2O3 films are nanocrystalline in nature with rhombohedral structure. The observed GIXRD data are in good agreement with the standard data [JCPDS: 79-0007 and JCPDS: 75-0765]. The (012), (104), (116) and (018) orientations observed in the pat- terns are due to a-Fe2O3, which is in good agreement with the results reported by several workers (Akhavan, 2010; An et al., 2009; Cha et al., 2009). For higher wt.% of Mn doping (045) and (120) peaks due to Mn3O4 are observed. It is also observed that the intensity of diffraction peaks due to Mn3O4 increases with doping percentage. The average crystal- lite size of the deposited material was determined by using Debye–Scherer formula (Ubale et al., 2013), 0:9k D ¼ ð3Þ bCosh where ‘k’ is the wavelength used (0.154 nm); ‘b’ is the angular line width at half maximum intensity in radians and ‘h’ is the Bragg’s angle. It is found that the crystallite size of the a-Fe2O3 film is of the order of 21 nm, and it decreases to 17 nm as doping percentage of Mn was increased from 0 to 8 wt.% (Table 1).

3.2. Surface morphology

The surface morphology of the iron oxide thin films deposited onto glass substrates by the SILAR method was examined by Figure 4 EDAX spectrum (A) as deposited Fe2O3 film and (B) using field emission scanning electron micrographs (Fig. 3). 6 wt.% Mn doped a-Fe2O3 film. 42 M.R. Belkhedkar et al.

Figure 5 3D AFM images of Mn doped a-Fe2O3 thin films; Mn doping percentage: (A) 0 wt.%, (B) 2 wt.%, (C) 4 wt.%, (D) 6 wt.% and (E) 8 wt.%.

Table 2 Morphological parameters of Mn doped a-Fe2O3 thin films. wt.% of Mn in RMS roughness Rq Average surface roughness Maximum height Average height Grain orientation

Fe2O3 (nm) Ra(nm) (nm) (nm) (pi) 0 86.01 71.83 323.2 121.72 0.07 2 77.53 62.14 267.5 118.01 0.16 4 71.05 59.82 280.6 117.34 0.08 6 68.72 57.45 317.4 114.3 0.08 8 66.15 54.03 326.6 112.78 0.12 Nanocrystalline Mn doped a-Fe2O3 thin film growth by SILAR method 43

3 range 350–750 nm at room temperature shown in the inset of 0.6 Fig. 6. An increase in the optical absorption of a-Fe2O3 films undoped Fe2O3 2.5 2% Mn doped Fe2O3 was observed with increase in doping percentage of Mn. The opti- 0.4 4% Mn doped Fe2O3 cal band gap energies (Eg) of undoped and Mn doped a-Fe2O3 6% Mn doped Fe2O3 films were calculated by using the equation (Mahdi et al., 2012), 2

t (cm) 8% Mn doped Fe2O3

2 α 0.2 aht ¼ Aðht EgÞn ð4Þ (eV/cm)

10 where, ‘a’ is absorption coefficient, ‘Eg’ is band gap energy, ‘A’ 1.5 0 10

× is a constant and ‘n’ is equal to 1/2 for direct and 2 for indirect

2 350 450 550 650 750 ) λ 2 ν Wavelength, (nm) transition. Fig. 6 illustrates the plots of (ahm) versus hm for Mn h α

( 1 doped a-Fe2O3 thin films. The band gap energy ‘Eg’ of the a- Fe2O3 film was estimated by extrapolating the linear portion of 0.5 the plot to the energy axis and is found to be of the order of 3.02 eV. This estimated optical band gap energy of the a- Fe2O3 thin film is in good agreement with the results reported 0 1.5 2 2.5 3 3.5 by Glasscock et al. (2008) and Bhar et al. (2010). However, it is hυ (eV) observed that, the optical band gap energy of the film decreases from 3.02 to 2.95 eV as Mn doping in a-Fe2O3 rises Figure 6 The plots of (ahm)2 versus hm (Inset plots of optical from 0 to 8 wt.%. absorption versus wavelength) of Mn doped a-Fe2O3 thin films. 3.4. Antibacterial activity

The antibacterial character of the Mn doped a-Fe2O3 thin films were investigated against S. aureus bacteria. The test results of S. aureus bacteria incubated at 24 h on undeposited glass substrate and Mn doped a-Fe2O3 thin film surfaces are shown in Fig. 7. It was observed that, the antibacterial effi- ciency for undoped a-Fe2O3 film is 16.66% and it increases to 58.33% as doping of Mn increases from 0 to 8 wt.%. This increased antibacterial efficiency may be due to the improved nanocrystalline nature of a-Fe2O3 films. But, the interactions of nanoparticles with bacteria are dependent on the size, shape and morphology of the deposited material (Yu et al., 2011; Sikong et al., 2010). When these nanoparticles come in contact with a bacterial cell, an active oxygen is formed due to the chemisorption process, as a result more number of iron, man- ganese and hydroxide ions or hydrogen peroxide were released from the surface, which can react with the peptide linkages in the cell wall of bacteria and disrupt them. This antibacterial mechanism may be involved in the antibacterial study to kill the S. aureus microorganisms (Zhang et al., 2008; Touati, 2000; Keenan and Sedlak, 2008; Ubale and Belkhedkar, 2015; Belkhedkar and Ubale, 2014; Aninwene et al., 2013). It was also observed from the AFM images that the rms rough- ness of a-Fe2O3 film decreases with the increase of Mn doping, which plays a very important role in the antibacterial activity against the S. aureus bacteria. Due to decreased rms roughness of the film, more nanoparticles of deposited material interacted with the bacteria and damaged the cell wall which further blocks the bacterial production. As the rms roughness of the a-Fe2O3 thin films is almost closure above 4 wt.% doping of Mn, its antibacterial activity against S. aureus bacteria becomes nearly constant. Similarly, the role of rms roughness on the antibacterial activity of nano-barium sulfate incorpo- rated pellethane bio-film against S. aureus and Pseudomonas aeruginosa bacteria was reported by Aninwene et al. (2013).

4. Conclusions Figure 7 Antibacterial test results of S. aureus after 24 h: incubated on (A) undeposited glass substrate and on Mn doped In the present work, Mn doped a-Fe2O3 thin films were suc- a-Fe2O3 thin films with Mn doping (B) 0 wt.%, (C) 2 wt.%, (D) cessfully synthesized by successive ionic layer adsorption and 4 wt.%, (E) 6 wt.% and (F) 8 wt.%. reaction method onto glass substrates. The GIXRD, 44 M.R. Belkhedkar et al.

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ORIGINAL ARTICLE Empirical models for estimating the mechanical and morphological properties of recycled low density polyethylene/snail shell bio-composites

C.U. Atuanya a, V.S. Aigbodion b,*, S.O. Obiorah a, M. Kchaou c,d, R. Elleuch d a Department of Metallurgical and Materials Engineering, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria b Department of Metallurgical and Materials Engineering, University of Nigeria, Nsukka, Nigeria c Department of Mechanical Engineering, Higher Institute of Applied Sciences and Technology, University of Sousse, Tunisia d Laboratory of Electromechanical Systems, National Engineering School of Sfax, University of Sfax, BP 3038, Tunisia

Received 5 April 2014; revised 4 December 2014; accepted 8 January 2015 Available online 21 February 2015

KEYWORDS Abstract The empirical models for estimating the mechanical properties and morphological of Polymer–matrix composites recycled low density polyethylene/snail shell bio-composites was investigated. The snail shell of par- (PMCs); ticle sizes 75, 125, 250 and 500 lm with a weight percentage of 0, 5, 10 and 15 (wt%) with recycled Mechanical properties; polyethylene (RLDPE) were prepared by compounding and compressive moulding technique. Sam- Microstructures; ples were cut from the panel and subjected to mechanical testing such as tensile, flexural and impact Statistical properties/meth- energy. Scanning electron microscope was used to analyse the fracture surface of the samples. ods and mechanical testing; Linear regression equation and analysis of variance (ANOVA) were employed to investigate the Electron microscopy influence of process parameters on the mechanical properties of the samples. Results obtained showed that: as the wt% snail shell particles increased from 5 to 15, there was a raise in the tensile strength by (2.69) and the flexural strength (1.53). Also the increase in the snail shell particle size from 75 to 500 lm decreased the tensile strength by 5.46, flexural strength 3.97 and impact energy by 1.97. The predicted results obtained were in good agreement with the experimental results. Hence, the work can be used for indoor and outdoor structural applications. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction development of composite materials. Polymer composites have received the attention of researchers because of low strength, Composites material have become important engineering hardness and wear of plastics or polymer for most engineering materials all over the world because of the unique properties applications. Polymer composites are now being used in both they offer when compared with polymer, metals or alloys. As indoor and outdoor structural applications in housing, con- a result most research and development are focusing on the struction, auto-industry, aerospace etc. Natural fillers in the form of fibres of particulate have * Corresponding author. Tel.: +234 8028433576. gained the attention of researchers in recent time as reinforcing E-mail address: [email protected] (V.S. Aigbodion). materials in polymers, metals and ceramics. They are eco- Peer review under responsibility of University of Bahrain. friendly, low cost, low density materials; they are renewable http://dx.doi.org/10.1016/j.jaubas.2015.01.001 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 46 C.U. Atuanya et al. in a large amount when compared with the artificial fillers. into a Denver cone crusher and was reduced to a size of 4– Some of the previous studies on the use of natural fillers on 3 mm wider and then charged into a roll crusher to reduce the production of polymer composites are: Patricio et al. the size of snail shells to between 2 and 1 mm. The products (2007) studied egg shell (ES) as a new bio-filler for polypropyl- from the roll crusher were transferred into a ball milling ene composites, water absorption and mechanical properties of machine and were left in the mill for 2 h; after which they were high – density polyethylene/egg shell composite were studied transferred into a set of sieves: 500 lm, 250 lm, 125 lm and by Hussein et al. (2011), the tribological behaviour of recycled 75 lm sizes and were sieved for 30 min using a sieve shaker low density polyethylene (RLDPE) polymer composites with machine. JOEL JSM 5900LV Scanning Electron Microscope bagasse ash particles as a reinforcement using a pin-on-disc equipped with an Oxford INCA Energy Dispersive Spectros- wear rig under dry sliding conditions was reported by copy system was used to determine the Snail shell particle Aigbodion et al. (2012a), Atuanya et al. (2011) investigated microstructure. the suitability of using recycled low density polyethylene The Snail shell particles were dried in oven (UNB 100–500) (RLDPE) in wood composite board manufacture, Aigbodion for 24 h at 150 C to a moisture content of 0.5–1% (based on et al. (2012b) studied high density polyethylene (HDPE) com- dry weight) before composite production (75 lm, 125 lm, posite reinforced with 20 wt% orange peels ash particles and 250 lm, 500 lm) and (0, 5, 10, 15 wt%) were compounded the effect of palm kernel shell on the microstructure and with recycled polyethylene using a co-rotating twin-screw mechanical properties of recycled polyethylene (RLDPE) was extruder. The temperature used was 160 C with screw speed reported by Agunsoye et al. (2012). of 50 rpm (Agunsoye et al., 2012). Finally, the mixed samples There are lots of snail shell and waste water sachet (recycled come out in bulk form. The crushing machine was used to low density polyethylene) waste materials, these wastes consti- crush into particle form. The composite compounded was fab- tute nuisance to the environment not only in Africa but the ricated into size of 150 mm · 150 mm · 6 mm. The used tem- world at large. The ability to convert these wastes into useful perature was 160 C with a pressure of 25 Ton for 10 min. engineering materials e.g. composites sharpens the focus of this Each of the plate was cut into desired dimension for sample present research work. From the available literature no inves- testing. Before the test, the samples were conditioned for tigation has been conducted on the application of the snail 24 h at 23 C and at 50% relative humidity (Suresha and shell particles in polymer composite materials. A relationship Chandramohan, 2004; Singha and Thakur, 2008). between the mechanical properties of the polymer composites The tensile test was determined in accordance with AST- and the process parameters (particle sizes and weight percent- MD638-10 Standard Test Method for Tensile Properties of age snail shell) will give a better understanding of the mechan- Plastics. This test method covers the determination of the ten- ical properties. Based on the above-mentioned situation, the sile strength of unreinforced and reinforced RLDPE in the study described in this work intends to study the empirical form of standard dumbbell-shaped test specimens. The sam- model for estimating mechanical properties of RLDPE com- ples were aligned properly to prevent bending moment occur posites reinforced with snail shell particles. during test. The cross-head speed was 5 mm/min and the gauge length was 40 mm. Three samples were tested for each compo- 2. Materials and method sition and a mean of these samples was taken for Young mod- ulus and tensile strength. Flexural strength of the three-point loading system applied 2.1. Materials to a simply supported beam was used. The samples were aligned properly to prevent error measurement during test. Pure water sachet (RLDPE) used were collected around the The flexural strength was determined in accordance with refused dump at the Faculty of Engineering, Nnamdi Azikiwe ASTM D790-10 Standard Test Methods for Flexural Proper- University, Awka, Anambra State, Nigeria. The RLDPE was ties of Unreinforced and Reinforced Plastics and Electrical washed, dried and pulverized to particles. The snail shell used Insulating Materials. The test was conducted with a cross-head in this work was brown snail shell obtained from Snail restau- speed of 5 mm/min and a gauge length of 100 mm. Three sam- rants in Sabongidda-Ora, Edo-State Nigeria. The Snail shell ples were tested for each composition and a mean of these sam- was washed with water and ethanol, sun dried to remove the ples was taken. residual organic matter. The Izod test method was used to study the sample impact energy in accordance with ASTMD256-10 Standard Test 2.2. Equipment Methods for determining the Izod Pendulum Impact Resis- tance of Plastics. The sample with dimension 55 · 10 · 4mm The equipment used for this research were: metal mould, was notched and placed vertically (Izod) in a vice with the sieves, digital weighing balance, hack saw grinding machine, notch positioned central to the top of the vice facing the swing hydraulic press, compounding machine, Compressive machine patch of the pendulum. The pendulum having a known energy, and housefield tensometer, scanning electron microscope strikes the sample and the swing height of the pendulum after (SEM). breaking the sample is measured and subtracted from the cal- ibrated swing height. The result is the absorbed energy which can break the sample. 2.3. Method A full factorial design of experiments of the type Pn (Miller and Freund, 2001) was used in the study of the mechanical 40 kg of the dried Snail shell were oven dried at 105 C for 5 h properties where n corresponds to the number of factors and until all the moisture was completely removed (Hassan et al., P represents the number of levels. Here i.e.: n corresponds to 2012; Atuanya et al., 2014). The dried Snail shell was charged the number of factors (Particle size and wt% snail shell Empirical models for recycled low density polyethylene/snail shell bio-composites 47 particles) and p the number of levels (P = 2) (upper and lower the abundant presence of calcium in the Snail shell particles. levels of each variable, see Table 1). Thus, the number of trial The ratio of calcium to oxygen was 2.3:1.8 this ratio was experiments to be conducted for each material is 4 (i.e. 22 = 4). attributed to also high level of oxygen because the Snail shell If the response variable is represented by Y, the linear regres- particles occurred in oxide form which was different from sion equation for these experiments is expressed as (Miller and the ratio of CaCO3. These results are consistent with the work Freund, 2001; Aigbodion et al., 2012a): of other co-workers that work on eggshell (Hussein et al., 2011; Patricio et al., 2007). Y ¼ a þ a A þ a B þ a AB ð1Þ ð1;2;3Þ 0 1 2 3 Fig. 2 shows the variation of tensile modulus and strength Where a0 is the response variable at the base level, a1, a2 are with wt% Snail shell particles. There was a significant incre- the coefficients associated with each variable A (particle size) ment in the tensile modulus and the strength as the snail shell and B (amount of snail shell), a3 the interaction coefficient between A and B within the selected levels of each variable and Y(1, 2, 3) represents tensile strength, flexural strength and impact energy respectively. The methodology for calculating the values for each regression coefficients, using the coded val- ues A and B of each variable is described elsewhere (Buggy et al., 2005).

3. Results and discussion

Surface Morphology of Snail shell particles is shown in Fig. 1a. Snail shell particles were clearly seen to be solid in nat- ure, but irregular in size. Some spherical shape particles and platy like structure can also be observed in Fig. 1a. The snail shell particles surface morphology plays a vital role in case of composite materials. The surface features of particles such as contours, defects and damage and surface layer were not observed in the SEM. The micro-analysis by EDS of the Snail shell particle morphology consists mainly of Ca, O, Si, Mg and Na elements (see Fig. 1b). The relative atomic percent of the atoms were obtained from the peak area and corrected with an appropriate sensitivity factor. The Snail shell particles showed a higher proportion of calcium atom. The higher proportion of calcium in the particles can be attributed to

Table 1 Upper and lower levels of each factor along with their coded values. S.No Variables Upper level Lower level A Particle size (lm) 500 (+1) 75 (1) B Amount of snail shell (wt%) 15 (+1) 5 (1) Figure 2 (a) Tensile modulus and (b) strength variation with wt% snail shell particles.

Figure 1 (a) SEM observation and (b) EDS analysis of the snail shell particles. 48 C.U. Atuanya et al. particle loading increased, smaller Snail shell particle sizes showed higher increments. The increment may be due to the platy structure of the Snail shell filler providing good reinforce- ment (Hassan et al., 2012; Hussein et al., 2011). The increase in modulus of the Snail shell-filled composites indicates an increase in the rigidity of RLDPE related to the restriction of the mobility in RLDPE matrix due to the pres- ence of snail shell particles. The modulus of these composites increased with increasing Snail shell particle loading. This sug- gests stress transfer across the polymer-particle interface. Snail shell in the matrix prevented movement in the area around each particle, contributing to an overall increase in the modu- lus. The high modulus values also support the use of the devel- oped composites in indoor and outdoor applications which was in par with the work of Hussein et al. 2011. Fig. 3 depicts the variation in flexural modulus and strength with wt% Snail shell particles. The flexural modulus and strength rise with the increment in wt% Snail shell particles. The rate of increase of flexural modulus and strength was comparable to the incre- ment in wt% Snail shell particles. The size of the Snail shell particles influenced the tensile and the flexural strengths (see Figs. 2 and 3). Snail shell particles were divided into coarse particles (around 500 lm), large par- ticles (around 125 and 250 lm) and fine particles (around 75 lm) according to their sizes. The mechanical interlocking between Snail shell particles and matrix (RLDPE) significantly affected the properties (Agunsoye et al., 2012; Bledzki and Gassan, 1999). The high values of strength observed in this work may be due to the fair distribution of the Snail shell particles in the RLDPE matrix resulting in strong particles–RLDPE matrix Figure 3 Variation of (a) the flexural modulus and (b) the interaction. The particle dispersion improved the particles– flexural strength with wt% snail shell particles. RLDPE matrix interaction and consequently increases the ability of the Snail shell particles to restrain gross deformation of the RLDPE matrix. The impact energy of a composite is influenced by many factors: including the toughness properties of the reinforce- ment, the nature of interfacial region and frictional work involved in pulling out the particles from the matrix (Atuanya et al., 2014). Fig. 4 depicts the variation in impact energy with wt% Snail shell particles. The Impact energy decreased with increment in the wt% Snail shell particles. This may be attributed to the interference by the filler in the mobil- ity or deformability of the matrix. This interference was cre- ated through the physical interaction and immobilization of the polymer matrix by imposing mechanical restraints (Hassan et al., 2012; Imoisili et al., 2013). It can be seen that the impact energy of the composites Figure 4 Variation of impact energy with wt% Snail shell slightly decreases with increasing filler loading. Increased filler particles. loading in the RLDPE matrix resulted in the stiffening and hardening of the composite. This reduced its resilience and toughness, and led to lower impact energy which is in par with variables corresponding to each set of trial are reported in the work of Atuanya et al.(2014) and Raju et al. (2012). As the Table 2. The respond variables in each trial represent the aver- loading of Snail shell particles increases, the ability of the com- age of three measured data at identical experimental posites to absorb impact energy decreases since there is less conditions. ratio of the RLDPE matrix to particles. The results obtained From the factorial design and the stepwise variation of the in this work are within the standard level for bio-composites two factors, the estimated response surfaces represents the for indoor and outdoor applications (Hussein et al., 2011). best fit of the experimentally obtained values. Figs. 5–7 show For the modelling of the mechanical properties, the upper the estimated response surface for the tensile strength, the level and the lower level of each variable along with their flexural strength and the impact energy as a function of coded values used in this investigation are shown in Table 1. particle size and wt% Snail shell particles. It can be seen that The design of the experiments and the values of respond both the tensile and the flexural strength were highly Empirical models for recycled low density polyethylene/snail shell bio-composites 49

Table 2 Matrix design for calculating the regression co-efficient and ANOVA. S.No ABABTensile strength (N/mm2) Flexural strength (N/mm2) Impact energy (J) S1 +1 +1 +1 16.71 25.70 3.71 S2 +1 1 1 11.5 18.50 7.30 S3 1 1 +1 22.26 25.10 10.50 S4 1+11 27.8 35.76 7.50 Note: +1 = upper level, 1 = lower level.

Tensile strength shell 500.00

14.1342 27.7175

23.6425 16.8508 393.75 19.5675

15.4925

19.5675 11.4175 287.50 Tensile strength A: Particles size

22.2842 181.25 15.00 500.00 25.0008 12.50 393.75 10.00 287.50 75.00 5.00 7.50 10.00 12.50 15.00 B: Amount of snail shell 7.50 181.25 A: Particles size 5.00 75.00 B: Amount of snail shell

Figure 5 Response variation of tensile strength with interaction between snail shell particle size and wt%.

Flexural strength 15.00

shell

35.475 12.50 33.6417 32.725

29.975 31.8083 29.975 28.1417 27.225 10.00

24.475 26.3083 Flexural strength B: Amount of snail shell 7.50

15.00 500.00 12.50 393.75 5.00 10.00 287.50 75.00 181.25 287.50 393.75 500.00 B: Amount of snail shell 7.50 181.25 A: Particles size A: Particles size 5.00 75.00

Figure 6 Response variation of flexural strength with interaction between snail shell particle size and wt%. influenced by the Snail shell particle content. For the Snail is highly influenced by the Snail shell particle content and shell particle size the tensile and the flexural strength were particle size. The impact energy was found to raise with a found to raise with decrement in the particle size from 500 decrease in the particle size (500–75 lm) and wt% snail shell to 75 lm. However, the estimated response surface indicates particles (15–5). However, the estimated response surface maximum composite strengths at 15 wt% Snail shell and indicates maximum composite impact energy at 5 wt% snail 75 lm particle size (see Figs. 5 and 6). The impact energy shell and 75 lm particle size (see Fig. 7). 50 C.U. Atuanya et al.

Impact Energy 15.00

shell 4.76417

10.4225 12.50

8.725 5.89583

7.0275

7.0275 5.33 10.00

3.6325 8.15917 Impact Energy B: Amount of snail shell

7.50 9.29083

15.00 500.00 12.50 393.75 5.00 10.00 287.50 75.00 181.25 287.50 393.75 500.00

B: Amount of snail shell 7.50 181.25 A: Particles size A: Particles size 5.00 75.00

Figure 7 Response variation of impact energy with interaction between snail shell particle size and wt%.

ANOVA was used to determine the design parameters sig- and B (wt% snail shell) were significant model terms. Values nificantly influencing the tensile strength, the flexural strength greater than 0.1000 indicate that the model terms are not sig- and the impact energy. Table 3 shows the results of ANOVA. nificant (see Table 3). The analysis was evaluated at confidence level of 95%, that is The mechanical properties such as tensile, flexural, and for significance level of a = 0.05 (Miller and Freund, 2001). impact strength were modelled using Design Expert statistical The last column of Table 3 shows the contribution (P) of each software. Eqs. (2)–(4) were the developed nonlinear regression parameter on the response, indicating the degree of influence models for tensile strength (Y1), flexural strength (Y2) and on the results. For the ANOVA of Tensile strength Test, the impact strength (Y3) respectively. fit was exact and the R2 value is 0.9971. The Model F-value Tensile strength ðY Þ¼þ19:57 5:46 A þ 2:69 B is 2722.61 implies, the model is significant, flexural strength 1 the R2 value is 0.9994. The Model F-value is 14501.00 implies, N=mm2 ð2Þ the model is significant and for impact energy the ANOVA was also fit, exact and the R2 value is 0.9838 with Model F- Flexural strength ðY2Þ¼þ29:97 3:97 A þ 1:53 B value of 492.34 which, also implies that the model is signifi- N=mm2 ð3Þ cant. Values of ‘‘Prob > F’’ less than 0.0500 indicate that the model terms are significant. In this case, A (particle size) Impact Energy ðY3Þ¼þ7:03 1:97 A 1:42 B J ð4Þ

Table 3 ANOVA for Selected Factorial Model.

Source Sum of squares Tensile strength Pvalue Remarks

DF Mean square Fvalue Model 148.25 2 74.12 2722.61 0.0136 Significant A 119.36 1 119.36 4384.04 0.0096 B 28.89 1 28.89 1061.18 0.0195 Residual 0.027 1 0.027 CorTotal 148.27 3 Flexural strength Model 72.50 2 36.25 14501.00 0.0059 Significant A 63.20 1 63.20 25281.00 0.0040 B 9.30 1 9.30 3721.00 0.0104 Residual 2.500E003 1 2.500E003 CorTotal 72.51 3 Impact energy Model 23.66 11.83 492.34 0.0319 Significant A 15.56 1 15.56 647.78 0.0250 B 8.09 1 8.09 336.90 0.0346 Residual 0.024 1 0.024 CorTotal 23.68 3 Empirical models for recycled low density polyethylene/snail shell bio-composites 51

Table 4 Validation of mathematical model. Std. Actual values Predicted values % of error TS (N/mm2) FS (N/mm2) IM (J) TS (N/mm2) FS (N/mm2) IM (J) TS (N/mm2) FS (N/mm2) IM (J) S1 22.26 32.40 10.50 22.34 32.47 10.42 0.36 0.22 0.76 S2 11.50 24.50 6.40 11.42 24.47 6.48 0.70 0.12 1.25 S3 27.80 35.50 7.50 27.72 35.47 7.58 0.35 0.08 1.07 S4 16.71 27.50 3.71 16.79 27.53 3.63 0.47 0.11 2.16 Average absolute error 0.47 0.13 1.85

a) Composite of 500 µm at 15 wt% snail shell, b) Composite of 75 µm at 15 wt% snail shell

Figure. 8 SEM tensile fracture surface of the composites.

Where (A) and (B) were the coded values of particle size and The fracture surfaces of the composites were examined wt% snail shell particles respectively. The value of a0 for ten- using JEOL JSM-6480LV scanning electron microscope. sile, flexural strengths and impact energy was 19.57, 29.97 and Fig. 8a shows the composite microstructure at 500 lm with 7.03 respectively. It represents the respond variable value at 15 wt% Snail shell particles. The microstructure showed that the base level. By substituting the coded values of the variables the Snail shell particles are embedded in the polymer matrix. for the experimental conditions in Eqs. (2)–(4), the tensile Snail shell particles are not broken and there were voids strength, the flexural strength and the impact energy for the around the particle indicating poor interaction. The SEM of composites can be calculated. It was noted that from Eqs. the fracture surface of the RLDPE and its composite at (2) and (3) that the coefficients of wt% Snail shell particles 15 wt% Snail shell particles at 75 lm is shown in Fig. 8b. Mor- (B) were found to be positive. It indicates that increase in phological properties’ result shows that there is proper inti- wt% Snail shell particles from 5 to 15 raises the tensile strength mate mixing of Snail shell particles with the RLDPE in the by (2.69) and the flexural strength by a factor of (1.53). Also, bio-composites synthesized. These factors are responsible for the coefficients of particle size (A) were found to be negative. It the increases in the results of the tensile, flexural strengths indicates that raise in particle size from 75 to 500 lm decreased and impact energy obtained in the composites with smaller the tensile strength by 5.46, the flexural strength by 3.97 particle size. and the impact energy by 1.97 (see Figs. 4–6). Confirmation experiments were conducted for four sets of conditions. The actual values and the predicted values obtained from the 4. Conclusions Regression model were compared (see Table 4). The percent- age of error was calculated using Eq. (5) for the validation The effects of Snail shell particles reinforced RLDPE compos- of the Regression model (Miller and Freund, 2001). ites have been investigated as a function of filler loading and particle size. Based on the results and discussion above the fol- % of error ¼ðActual value lowing conclusions can be made: The incorporation of the Snail shell particles in the RLDPE polymer matrix as a rein- Predicted valueÞ=Actual value 100% ð5Þ forcement increases the tensile and the flexural strength of From Table 4, the averages absolute error for the tensile the material. Increases in wt% Snail shell particles from 5 to strength, the flexural strength and the impact energy are found 15 raises the tensile strength and the flexural strength by a fac- to be 0.47, 0.13 and 1.85% respectively, which means that a tor of 2.69 and 1.53 respectively. Increases in Snail shell parti- better accuracy was obtained using the developed Regression cle size from 75 to 500 lm decreased the tensile strength by models. 5.46, the flexural strength by 3.97 and the impact energy 52 C.U. Atuanya et al. by 1.97. Factorial design of the experiment can be success- Bledzki, A.K., Gassan, J., 1999. Composites reinforced with cellulose fully employed to describe the mechanical properties of the based fibres. Prog. Polym. Sci. 24, 221–274. samples and the developed linear equation models can be used Buggy, M., Bradley, G., Sullivan, A., 2005. Polymer filled interaction in predicting the mechanical behaviour of the materials within in kaoline/nylon 6.6 composites containing a silane coupling agent. Composites Part A 36, 437–442. the selected experimental conditions. Based on the results Hassan, S.B., Oghenevweta, J.E., Aigbodion, V.S., 2012. Morpholog- obtained in this study, it is recommended that the composites ical and mechanical properties of carbonized waste maize stalk as can be used in the production of indoor and outdoor reinforcement for eco-composites. Composites Part B 43, 2230– applications. 2236. Hussein, Abdullah A., Salim, Rusel D., Sultan, Abdulwahab A., 2011. References Water absorption and mechanical properties of high-density polyethylene/egg shell composite. J. Basrah Res. (Sci.) 37, 36–42. Agunsoye, J.O., Talabi, S.I., Obe, A.A., Adamson, I.O., 2012. Effects Imoisili, P.E., Ezenwafor, T.C., AttahDaniel, B.E., Olusunle, S.O.O., of palm kernel shell on the microstructure and mechanical 2013. Mechanical properties of Cocoa-Pod/Epoxy composite; effect properties of recycled polyethylene/palm kernel shell particulate of filler fraction. Am. Chem. Sci. J. 3 (4), 526–531. composites. J. Mineral. Mater. Charact. Eng. 11, 825–831. Miller, I., Freund, J.E., 2001. Probability and Statistics for Engineers. Aigbodion, V.S., Hassan, S.B., Agunsoye, O.J., 2012a. Effect of Prentice Hall India Ltd, India, pp. 125–140. bagasse ash reinforcement on dry sliding wear behaviour of Patricio, T., Rau´l, Q., Mehrdad, Y.P., Jose´, L.A., 2007. Eggshell, a polymer matrix composites. Mater. Des. 2012 (33), 322–327. new bio-filler for polypropylene composites. Mater. Lett. 61, 567– Aigbodion, V.S., Hassan, S.B., Atuanya, C.U., 2012b. Kinetics of 572. isothermal degradation studies by thermogravimetric data: effect of Raju, G.U., Gaitonde, V.N., Kumarappa, S., 2012. Experimental orange peels ash on thermal properties of high density polyethylene study on optimization of thermal properties of groundnut shell (HDPE). J. Mater. Environ. Sci. 3 (6), 1027–1036. particle reinforced polymer composites. Int. J. Emerg. Sci. 2 (3), Atuanya, C.U., Ibhadode, A.O.A., Igboanugo, A.C., 2011. Potential 433–454. of using recycled low- density polyethylene in wood composites Singha, A.S., Thakur, Vijay Kumar, 2008. Mechanical properties of board. Afr. J. Environ. Sci. Technol. 5, 389–396. natural fibre reinforced polymer composites. Bull. Mater. Sci. 31 Atuanya, C.U., Edokpia, R.O., Aigbodion, V.S., 2014. The physio- (5), 791–799, ª Indian Academy of Sciences. mechanical properties of recycled low density polyethylene Suresha, B., Chandramohan, G., 2004. Friction and wear character- (RLDPE)/bean pod ash particulate composites. Results Phys. 4 istics of carbon-epoxy and glass-epoxy oven roving fiber compos- (2014), 88–95. ites. J. Reinf. Plast. Compos. 64, 67–72. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016) 21,53–58

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ORIGINAL ARTICLE Thermodynamic properties and approximate solutions of the ‘-state Po¨schl–Teller-type potential

W.A. Yahya a,b,*, K.J. Oyewumi a a Theoretical Physics Section, Department of Physics, University of Ilorin, Nigeria b Department of Physics and Material Science, Kwara State University, Malete, Nigeria

Received 2 January 2015; revised 10 March 2015; accepted 2 April 2015 Available online 23 April 2015

KEYWORDS Abstract In this study, the solutions of the ‘-state Po¨schl–Teller-type potential for the Schro¨dinger Po¨schl–Teller-type potential; and Klein–Gordon equations are obtained using the parametric Nikiforov–Uvarov method. Klein–Gordon equation; Solving the Schro¨dinger and Klein–Gordon wave equations, the energy eigenvalues and wave func- Schro¨dinger equation; tions are obtained. For the case ‘ ¼ 0, we made comparison with previous results where the solu- Thermodynamic properties; tions of Schro¨dinger equation for the Po¨schl–Teller-type potential were obtained for s-wave (‘ ¼ 0) Parametric Nikiforov– state. We also obtain the thermodynamic properties such as vibrational mean energy, vibrational Uvarov method specific heat, vibrational mean free energy and vibrational entropy for the Po¨schl–Teller-type poten- tial in the classical limit. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction mechanics (Hassanabadi et al., 2012; Oyewumi and Akoshile, 2010), asymptotic iteration method (Champion The exact solutions of the wave equations in non-relativistic et al., 2008; Fernandez, 2004), improved AIM (Boztosun and relativistic quantum mechanics are very important. The and Karakoc, 2007; Yahya et al., 2014a), Laplace integral Schro¨dinger wave equation is used to describe non-relativis- transform (Ortakaya, 2012), factorization method (Dong tic spinless particles. The Klein–Gordon, Dirac, and Duffin– et al., 2007), proper quantization rule and exact quantization Kemmer–Petiau equations are used to describe spin zero, rule (Dong and Gonzalez-Cisneros, 2008; Qiang and Dong, spin half and spin one particles, respectively. The Duffin– 2010). The results obtained by solving wave equations for Kemmer–Petiau equation can also be used to describe spin certain potential models are increasingly being applied. zero particles. To obtain the exact and approximate solu- Recently, the solution of the two-dimensional spinless tions of the wave equations, various methods have been Klein–Gordon equation for scalar–vector harmonic oscilla- used ranging from Nikiforov–Uvarov method (Ikhdair, tor potentials with and without the constant perpendicular 2012; Yahya et al., 2010), supersymmetry quantum magnetic and Aharonov–Bohm (AB) flux fields was studied by Ikhdair and Falaye (2014). The energies and wave func- tions of certain potential models have also been used to * Corresponding author at: Department of Physics and Material obtain information-theoretic measures such as Fisher infor- Science, Kwara State University, Malete, Nigeria. mation, Shannon entropy, Renyi entropy, Tsallis entropy E-mail addresses: [email protected] (W.A. Yahya), mjpysics@ yahoo.com (K.J. Oyewumi). among other information-theoretic measures (see e.g Yahya et al., 2014b). Peer review under responsibility of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2015.04.001 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 54 W.A. Yahya, K.J. Oyewumi

ða;bÞ In this work, the solutions of Schro¨dinger and Klein– where Pn ðxÞ is the Jacobi polynomials and Gordon equations for the Po¨schl–Teller-type potential are 1 1 2 obtained for all ‘ (orbital angular momentum). The Po¨schl– c4 ¼ ð1 c1Þ; c5 ¼ ðc2 2c3Þ; c6 ¼ c5 þ A; c7 ¼ 2c4c5 B 2 2 ffiffiffiffi Teller potential is used to account for the Physics of many sys- 2 p c8 ¼ c þ C; c9 ¼ c3ðc7 þ c3c8Þþc6; c10 ¼ c1 þ 2c4 þ 2 c8 tems which includes the excitons, quantum wires and quantum 4 pffiffiffiffi pffiffiffiffi pffiffiffiffi dots (Ikhdair and Falaye, 2013a). The dynamical group of the c11 ¼ c2 2c5 þ 2ð c9 þ c3 c8Þ; c12 ¼ c4 þ c8; pffiffiffiffi pffiffiffiffi modified Po¨schl–Teller potential was studied by Dong and c13 ¼ c5 ð c9 þ c3 c8Þð5Þ Lemus in 2002, and it was realized as SU(1,1) group by factor- In the special case c3 ¼ 0, we have ization method (Dong and Lemus, 2002). Also the solutions of  the Dirac equation with the generalized Poschl–Teller poten- ¨ c11 c101; c101 c3 tial including the pseudo-spin-centrifugal term have been c101 limPn ð1 2c3sÞ¼Ln ðc11sÞ; ð6Þ obtained by Jia et al. (2009). The Po¨schl–Teller-type potential c3!0 to be considered is given as (Chen et al., 2013) c13 c12 c s c3 13 2 2 limð1 2c3sÞ ¼ e ; ð7Þ h a kðk þ 1Þ c3!0 VðrÞ¼ tanh2ðarÞ; ð1 < r < 1Þ; ð1Þ 2M and the wave function in Eq. (4) turns to where M is the mass of the particle, k denotes the potential W sc12 ec13sLc101 c s ; 8 depth and a is related with the range of the potential. ¼ n ð 11 Þ ð Þ We also study the thermodynamic properties such as vibra- a where LnðxÞ is the Laguerre polynomials. tional mean energy, vibrational specific heat, vibrational mean free energy and vibrational entropy for the Po¨schl–Teller-type 3. Solution of the Po¨schl–Teller-type potential potential. Thermodynamic properties of some model poten- tials were investigated recently. In Ref. (Baria and Jani, 3.1. Solution of the Schro¨dinger equation 2012), a new model potential was used with the exchange and correlation effects to calculate internal energy (enthalpy), entropy and Helmholtz free energy of liquid Na, K, Rb and Cs The radial part of Schro¨dinger equation in spherical polar at various temperatures with the variational approach. The coordinate can be written as  thermodynamic properties have also been studied for the 2 2 d Rn‘ðrÞ 2l h ‘ð‘ þ 1Þ modified Rosen–Morse potential (Dong and Cruz-Irisson, 2 þ 2 En‘ VðrÞ Rn‘ðrÞ¼0; ð9Þ dr h 2lr2 2012), harmonic oscillator plus an inverse square term (Dong et al., 2007), shifted Deng–Fan potential (Oyewumi et al., where l is the mass of the particle and En‘ is the energy spec- 2013) and Po¨schl–Teller potential (Ikhdair and Falaye, trum. The exact solution of Eq. (9) cannot be obtained except 2013a) which is of course different from the Po¨schl–Teller-type by using an approximation. It is found that the following potential to be considered in this study. approximation (Ikhdair and Falaye, 2013a, 2014; Ikhdair The paper is organized as follows: In Section 2, the para- and Hamzavi, 2012)  metric NU method will be reviewed. The bound state solutions 1 2 1 for the Po¨schl–Teller-type potential are obtained for a 4d0 þ ð10Þ r2 sinh2ðarÞ Schro¨dinger and Klein–Gordon equations using the paramet- ric Nikiforov–Uvarov method in Section 3. In Section 4, the is a good one to the centrifugal term in short range potential, thermodynamic properties such as vibrational mean energy, with d0 ¼ 1=12. The approximation used in Eq.ÀÁ(10) above is a 2 2 2 vibrational specific heat, vibrational mean free energy and slightly better approximation than 1=r a d0 þ 1=sinh ðarÞ vibrational entropy are studied for the Po¨schl–Teller-type at certain small values of ar like when a ¼ 0:1. Substituting potential. The conclusion is given in Section 5. Eqs. (1) and (10) into Eq. (9), we obtain 2 2 2. The parametric Nikiforov–Uvarov (NU) method d Rn‘ðrÞ a ‘ð‘ þ 1Þ 2 2 þ a kðk þ 1Þtanh ðarÞ Rn‘ðrÞ¼0; ð11Þ dr2 sinh2ðarÞ By using the parametric NU method, the solutions of a second where order differential equation of the form (Tezcan and Sever, 2009) "# 2lE 2 2 2 ¼ 4d0a ‘ð‘ þ 1Þ: ð12Þ d W c1 c2s dW As þ Bs C h2 2 þ þ 2 W ¼ 0 ð2Þ ds sð1 c3sÞ ds s2ð1 c sÞ 3 Also, if we make the substitution s ¼ sinh2ðarÞ, we obtain are 2 pffiffiffiffi pffiffiffiffi d Rn‘ðsÞ 1=2 þ s dRn‘ðsÞ 1 c n ð2n þ 1Þc þð2n þ 1ÞðÞþc þ c c nðn 1Þc þ þ 2 5 9 3 8 3 ds2 sð1 þ sÞ ds 2 2 pffiffiffiffiffiffiffiffi s ð1 þ sÞ þ c7 þ 2c3c8 þ 2 c8c9 ð sÞ ð KÞ K s2 þ s R ðsÞ¼0; ð13Þ ¼ 0 ð3Þ 4 4 4 n‘ and where  c11 2 2 c c101; c101 K ‘ ‘ 1 ; 1 : 14 13 c3 ¼ a ð þ Þ s ¼ a kðk þ Þ ð Þ c12 c12 c Wn ¼ s ðÞ1 c3s 3 Pn ð1 2c3sÞ; ð4Þ Comparing Eq. (13) with Eq. (2), we have that Thermodynamic properties of the ‘-state Po¨schl–Teller-type potential 55

s K K A ¼ ; B ¼ ; C ¼ 4 4 4 1 c ¼ ; c ¼1; c ¼1: ð15Þ 1 2 2 3 If we make use of Eq. (5), we obtain 1 1 1 s 1 K c ¼ ; c ¼ ; c ¼ þ ; c ¼ ; 4 4 5 2 6 4 4 7 4 4 1 K 1 s c ¼ þ ; c ¼ þ ; c ¼ 1 þ 2f; 8 16 4 9 16 4 10 1 1 c ¼2 þ 2ðc fÞ; c ¼ þ f; c ¼ ðc fÞ; ð16Þ 11 12 4 13 2 where rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s 1 K c ¼ þ ; f ¼ þ : ð17Þ 16 4 16 4 Substituting Eq. (16) into Eq. (3), we obtain after Figure 1 Energy eigenvalues of the non-relativistic Po¨schl–Teller simplification potential against n for various values of ‘ with a ¼ 1. 3 ¼4n2 4n þ 8nc þ 4c 8nf 4f K þ 8cf; ð18Þ 2 from which we obtain the energy eigenvalues as  d2 ‘ð‘þ1Þ 1 2 þ ½E2 M2c4 ½EþMc2VðrÞ uðrÞ¼0; 2 2 2 2 2 2 2 h 2 2 3 dr r h c h c En‘ ¼ 4d0a ‘ð‘ þ 1Þ4n 4n þ 8nc þ 4c 8nf 4f K þ 8cf ; 2l 2 ð21Þ ð19Þ where E refers to the energy spectrum and M is the rest mass of where n ¼ 0; 1; 2; ...; ½k and ½k denotes the largest integer the confined particle. If we make use of Eqs. (1) and (10),we inferior to k. Also, substituting Eq. (16) into Eq. (4), we obtain obtain the wave function as 2 d uðrÞ K 2 1 1c þ e jtanh ar uðrÞ¼0; ð22Þ 4þf 4 ð2f;2cÞ 2 2 Rn‘ ¼ s ð1 þ sÞ Pn ð1 þ 2sÞ: ð20Þ dr sinh ar The numerical results of the energy eigenvalues for the non- where relativistic Po¨schl–Teller-type potential are obtained in E2 M2c4 ðE þ Mc2Þ Table 1 for ‘ ¼ 0 (s-wave), and compared with the result 2 2 e ¼ 2 4a d0‘ð‘ þ 1Þ; j ¼ 2 a kðk þ 1Þ: ð23Þ obtained in Ref. (Chen et al., 2013) where the s-wave state of h c2 Mc the Po¨schl–Teller-type potential was studied. It can be Substituting s ¼ sinh2ðarÞ in Eq. (22), we obtain observed that our results are in good agreement. It can also d2uðsÞ 1=2 þ s duðsÞ 1 be noticed from Fig. 1 that the energy eigenvalue increases þ þ ds2 sð1 þ sÞ ds 2 2 with increasing n and ‘. s ð1 þ sÞ ðe jÞ ðe KÞ K s2 þ s uðsÞ¼0: ð24Þ 3.2. Solutions of the Klein–Gordon equation 4 4 4 Comparing Eq. (24) with Eq. (2), we have that The radial part of time-independent Klein–Gordon equation with equal scalar SðrÞ and vector VðrÞ potentials, describing j e e K K A ¼ ; B ¼ ; C ¼ spin-zero particle can be written as 4 4 4 1 c ¼ ; c ¼1; c ¼1: ð25Þ 1 2 2 3 Table 1 Energy eigenvalues for the non-relativistic Po¨schl– Using Eqs. (5) and (25), we obtain Teller-type potential with k ¼ 50; h ¼ 2l ¼ 1; a ¼ 1;‘¼ 0. 1 1 1 j e 1 e K nEn (our result) Enþ1 (Ref. Chen et al., 2013) c ¼ ; c ¼ ; c ¼ þ ; c ¼ ; 4 4 5 2 6 4 4 7 4 4 0 149.00 149.00 1 K 1 j 1 341.00 341.00 c8 ¼ þ ; c9 ¼ þ ; c10 ¼ 1 þ 2f; 2 525.00 525.00 16 4 16 4 3 701.00 701.00 1 1 c11 ¼2 þ 2ðd fÞ; c12 ¼ þ f; c13 ¼ ðd fÞ; ð26Þ 4 869.00 869.00 4 2 5 1029.0 1029.0 where 6 1181.0 1181.0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 1325.0 1325.0 j 1 K 1 d ¼ þ ; f ¼ þ : ð27Þ 8 1461.0 1461.0 4 16 4 16 9 1589.0 1589.0 10 1709.0 1709.0 Substituting Eq. (26) into Eq. (3), we obtain after simplification 56 W.A. Yahya, K.J. Oyewumi

3 (1) The vibrational mean energy U: e ¼4n2 4n þ 8nd þ 4d 8nf 4f K þ 8df; ð28Þ 2 @ UðbÞ¼ ln ZðbÞð34Þ from which we obtain the relativistic energy eigenvalues as @b

1 8 sffiffiffiffiffiffiffiffiffiffi 2 2 2 2 3 2 2 4 < rffiffiffiffiffiffiffiffiffiffi  En‘ ¼ h c 4n 4nþ8ndþ4d8nf4f Kþ8dfþ4a d0‘ð‘þ1Þ þM c : 2 2 2 2 2 2 1 J 8h b lp J hQ h bL hL h bQ UðbÞ¼pffiffiffi e e pffiffiffiffiffiffiffiffiffiffiffie 2l pffiffiffiffiffiffiffiffiffiffiffi e 2l ð29Þ lX : p 8h2b 2lpb 2lpb sffiffiffiffiffiffiffiffi 39 The wave function is also obtained as rffiffiffiffiffiffi = 1 lp 1 h2p J ffiffiffiffiffiffi J 5 1 e X p e !X ; ð35Þ 1 f d 2f; 2d 3=2 4þ 4 ð Þ 2 24 lb 2 ; un‘ ¼ s ð1 þ sÞ Pn ð1 þ 2sÞ: ð30Þ 4hb The numerical results of the eigenvalues for the relativistic where Po¨schl–Teller-type potential are displayed in Table 2 for some ! ¼ 2‘ð‘ þ 1Þa2 þ 3ð1 þ 8c2 þ 8f2 2KÞ; values of n and ‘. It is observed that when ‘(n) is kept constant, sffiffiffiffiffiffi ! sffiffiffiffiffiffi ! the energy eigenvalue increases with increasing n(‘). b b X ¼ erfi h Q erfi h L : ð36Þ 2l 2l 4. Thermodynamic properties (2) Vibrational specific heat C: To study the thermodynamic properties of the Po¨schl–Teller- @U @U type potential, we first obtain the vibrational partition func- CðbÞ¼ ¼kb2 : ð37Þ tion defined as @T @b 8 sffiffiffiffiffiffiffiffiffiffi 2 sffiffiffiffiffiffiffiffi 39 X½k 2 < 2 rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = bEn kb J 2bh 4 lp J lp J 1 h p J 5 ZðbÞ¼ e ; ð31Þ CðbÞ¼pffiffiffi 2e U e U e X e !X lX2 : p 8h2b 32h2b3 24 2lb ; n¼0 8 sffiffiffiffiffiffiffiffi2 sffiffiffiffiffiffiffiffi 39 3=2 < 2 rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = where b 1=kT; k is the Boltzmann constant and k is the kb J 2h 4 lp J lp J 1 h p J 5 ¼ ½ þ pffiffiffi e e U e X e !X largest integer inferior to k, the potential depth. The principal lX : p 8h2b 32h2b3 24 2lb ; 8 2 sffiffiffiffiffiffiffiffi 39 n ; ; ; ... 2 < rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = quantum number ranges from 0 1 2 ½k. In the classical kb pffiffiffi lp lp 1 h p þ pffiffiffiffiffiffiffiffiffiffiffiffiffi eJh3 b!4 eJU eJX eJ!X5 limit, at high temperature T for large ½k, the sum can be 18l3pX : 8h2b 32h2b3 24 2lb ; 8 sffiffiffiffiffiffiffiffiffiffi replaced by an integral and ½k can be replaced by k (since " rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi 2 < 2 h2 bL2 =2l k k 1). By substituting Eq. (19) into Eq. (31) and replac- kb J 8bh 1 J lp lp J e hL ½ ¼ þ pffiffiffi e e U þ e pffiffiffiffiffiffiffiffiffiffiffiffiffi ing the sum by an integral, the partition function for the lX : p 2hb3=2 2 8h2b 8lpb3 ! sffiffiffiffiffiffiffiffi Po¨schl–Teller-type potential gives, for large k: eh2 bL2 =2lh3L3 eh2 bQ2 =2lhQ eh2 bQ2 =2lh3Q3 1 h2p "# sffiffiffiffiffiffi ! sffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffi eJU! Z rffiffiffiffiffiffiffiffiffiffiffi 3 3 3 k 8l pb 8lpb 8l pb 12 2lb mp b b sffiffiffiffiffiffiffiffiffiffi bEn J rffiffiffiffiffiffi #) ZðbÞ¼ e dn ¼ 2e erfi h Q erfi h L ; ð32Þ rffiffiffiffiffiffiffiffiffiffiffiffi 2 3 0 8bh 2l 2l 9 lp 1 h p h p þ eJX þ eJ!X þ eJ!2X ; ð38Þ 8 2h2b5 24 2lb3 288l3=2 2b where where 2 ÂÃ h b 2 2 2 J ¼ 2‘ð‘ þ 1Þa þ 3ð1 þ 8c þ 8f 2KÞ ; hQ h2bL2 hL h2bQ2 12l U ¼ pffiffiffiffiffiffiffiffiffiffiffi e 2l pffiffiffiffiffiffiffiffiffiffiffi e 2l : ð39Þ 2lpb 2lpb L ¼ 1 2c þ 2f; Q ¼ 1 2c þ 2f þ 2k; Z z (3) Vibrational mean free energy F: erfðizÞ 2 2 erfiðzÞ¼ ; erfðzÞ¼pffiffiffi et dt: ð33Þ i p 0 FðbÞ¼kT ln ZðbÞð40Þ The thermodynamic properties can now be obtained from the ()rffiffiffiffiffiffiffiffiffiffiffi "# sffiffiffiffiffiffi ! sffiffiffiffiffiffi ! 1 mp b b partition function as follows: FðbÞ¼ ln eJ Erfi h Q Erfi h L : b 8bh2 2l 2l Table 2 Energy eigenvalues for the relativistic Po¨schl–Teller- ð41Þ type potential with k ¼ 50; h ¼ 2M ¼ c ¼ 1; a ¼ 1. (4) Vibrational entropy S: ‘ nEn‘ @ @ 000.488561, 7.66370 SðbÞ¼k ln ZðbÞþkT ln ZðbÞ¼k ln ZðbÞkb ln ZðbÞ: @T @b 010.440421, 13.1143 020.352923, 17.3663 ð42Þ 030.226206, 20.9756 ()rffiffiffiffiffiffiffiffiffiffi "# sffiffiffiffiffiffi ! sffiffiffiffiffiffi ! 110.439820, 13.1606 mp b b SðbÞ¼kln eJ Erfi h Q Erfi h L 120.351936, 17.4046 8bh2 2l 2l 8 sffiffiffiffiffiffiffiffiffiffi 130.224834, 21.0089 < rffiffiffiffiffiffiffiffiffiffi  2 2 2 2 2 1 J 8h b lp J hQ h bL hL h bQ 140.058249, 24.1800 ffiffiffi e e pffiffiffiffiffiffiffiffiffiffiffi e 2l pffiffiffiffiffiffiffiffiffiffiffi e 2l p : 2 220.350064, 17.4771 lTX p 8h b 2lpb 2lpb sffiffiffiffiffiffiffiffi 39 230.222230, 21.0717 rffiffiffiffiffiffi = 1 lp 1 h2p 240.054909, 24.2360 eJX pffiffiffiffiffiffi eJ !X5 ð43Þ 3=2 2 24 lb 2 ; 2 5 0.152284, 27.0754 4hb Thermodynamic properties of the ‘-state Po¨schl–Teller-type potential 57

The plots of the thermodynamics properties against k for the 1.4 diatomic molecules HCl and H2 with b ¼ 0:001 are shown in Figs. 2–6. The spectroscopic constants of the diatomic 1.35 molecules studied in this work are displayed in Table 3 and 1.3 HCl taken from (Yahya et al., 2014a). We have also used the H 1 2 conversions hc ¼ 1973:29 eV A˚ and 1 amu ¼ 931:494028 1.25 106 eV c2. It is observed from Fig. 2 that the partition func- tion Z decreases monotonically with increasing k for the two C 1.2 diatomic molecules considered, and reaches a constant value 1.15 for some typical values of k. From the variations of the mean energy U with k in Fig. 3, it can be observed that the vibra- 1.1 tional mean energy initially decreased to a minimum after which it increases with increasing k for the two diatomic 1.05 molecules studied. Fig. 4 shows that the vibrational specific 1 heat decreases exponentially with increasing k unlike the 1000 1500 2000 2500 3000 3500 4000 4500 5000 vibrational mean free energy F that increases monotonically λ with increasing k for the two diatomic molecules, as depicted Figure 4 Vibrational specific heat C against k for the diatomic in Fig. 5.InFig. 6, the variations of the vibrational entropy molecules HCl and H with b ¼ 0:001. S with k are shown. It is observed that the vibrational entropy 2 decreases monotonically with increasing k.

70 −1500

60 −2000 HCl H 2 HCl 50 H −2500 2

40

F −3000 Z

30 −3500

20 −4000

10

−4500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 λ 1000 1500 2000 2500 3000 3500 4000 4500 5000 λ Figure 5 Vibrational mean energy F against k for the diatomic

Figure 2 Vibrational partition function Z against k for the molecules HCl and H2 with b ¼ 0:001. diatomic molecules HCl and H2 with b ¼ 0:001.

5.5 1140

1130 5 1120 HCl HCl H H 2 1110 2 4.5

1100

S 4

U 1090

1080 3.5 1070

1060 3 1050

1040 2.5 1000 1500 2000 2500 3000 3500 4000 4500 5000 1000 1500 2000 2500 3000 3500 4000 4500 5000 λ λ

Figure 3 Vibrational mean energy U against k for the diatomic Figure 6 Vibrational entropy S against k for the diatomic molecules HCl and H2 with b ¼ 0:001. molecules HCl and H2 with b ¼ 0:001. 58 W.A. Yahya, K.J. Oyewumi

Dong, S.H., Lozada-Cassou, M., Yu, J., Jimenez-Angeles, F., Rivera, Table 3 Spectroscopic constants of the diatomic molecules A.L., 2007. Hidden symmetries and thermodynamic properties for studied in this work. a harmonic oscillator plus an inverse square potential. Int. J. Molecule l (amu) a (A˚1) Quant. Chem. 102 (2), 366. Fernandez, F.M., 2004. On an iteration method for eigenvalue HCl 0.9801045 1.8677 problems. J. Phys. A: Math. Gen. 37, 6173. H2 0.5039100 1.9425 Hassanabadi, H., Maghsoodi, E., Zarrinkamar, S., 2012. Relativistic symmetries of Dirac equation and the Tietz potential. Eur. Phys. J. Plus. 127 (3), 1. Ikhdair, S.M., 2012. Approximate-state solutions to the Dirac– 5. Conclusion Yukawa problem based on the spin and pseudospin symmetry. Cent. Eur. J. Phys. 10, 361. The solutions of the Po¨schl–Teller-type potential for the Ikhdair, S.M., Falaye, B.J., 2013a. Approximate analytical solutions Schro¨dinger and Klein–Gordon equations have been obtained to relativistic and nonrelativistic Po¨schl–Teller potential with its via the parametric Nikiforov–Uvarov method. We made com- thermodynamic properties. Chem. Phys. 421, 84. Ikhdair, S.M., Falaye, B.J., 2014. A charged spinless particle in parison between the energy eigenvalues obtained for the scalar–vector harmonic oscillators with uniform magnetic and Schro¨dinger equation (when ‘ ¼ 0) and that of the result Aharonov–Bohm flux fields. J. Assoc. Arab. Univ. Basic Appl. Sci. obtained in the literature where s-wave (‘ ¼ 0) state of the 16, 1–10. Po¨schl–Teller-type potential was considered in the non-rela- Ikhdair, S., Hamzavi, M., 2012. Approximate Dirac solutions of a tivistic case. The results are in perfect agreement. We have also complex parity-time-symmetric Po¨schl–Teller potential in view of obtained, from the high temperature partition function, the spin and pseudospin symmetries. Phys. Scr. 86, 045002. thermodynamic properties such as vibrational mean energy, Jia, C.S., Chen, T., Cui, L., 2009. Approximate analytic solutions of vibrational specific heat, vibrational mean free energy and the Dirac equation with the generalized Poschl–Teller potential vibrational entropy for the model potential. From the plots including the pseudo-spin-centrifugal term. Phys. Lett. A 373, 1621. of the various thermodynamic properties with k, we have Ortakaya, S., 2012. Exact solutions of the Klein–Gordon equation with ring-shaped oscillator potential by using the Laplace integral observed that the vibrational entropy and vibrational specific transform. Chin. Phys. B 21 (7), 070303. heat decrease with increasing k while the vibrational free Oyewumi, K.J., Akoshile, C.O., 2010. Bound-state solutions of the energy F increases monotonically with increasing k. The mean Dirac–Rosen–Morse potential with spin and pseudospin symmetry. energy U on the other hand initially decreases to a minimum Eur. Phys. J., A 45, 311. after which it continues to rise with increase in k. Oyewumi, K.J., Falaye, B.J., Onate, C.A., Oluwadare, O.J., Yahya, W.A., 2013. Thermodynamic properties and the approximate References solutions of the Schro¨dinger equation with the shifted Deng–Fan potential model. Mol. Phys. 112 (1), 127. http://dx.doi.org/10.1080/ 00268976.2013.804960. Baria, J.K., Jani, A.R., 2012. Thermodynamics of liquid alkalimetals Qiang, W.C., Dong, S.H., 2010. Proper quantization rule. Eur. Phys. using pseudopotential perturbation Scheme. Turk. J. Phys. 36, 179. Lett. 89, 10003, solved by the exact quantization rule. J. Phys. A: Boztosun, I., Karakoc, M., 2007. An improvement of the asymptotic Math Theor. 42, 035303. iteration method for exactly solvable eigenvalue problems. Chin. Tezcan, C., Sever, R., 2009. 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ORIGINAL ARTICLE A fractional model of fluid flow through porous media with mean capillary pressure

Anupama Choudhary a,*, Devendra Kumar b, Jagdev Singh c a Department of Mathematics, Arya College of Engineering & Research Centre, Jaipur 302028, Rajasthan, India b Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India c Department of Mathematics, Jagan Nath University, Jaipur 303901, Rajasthan, India

Received 16 June 2014; revised 29 December 2014; accepted 22 January 2015 Available online 7 March 2015

KEYWORDS Abstract In this paper, we discuss a fractional model arising in flow of two incompatible liquids Fluid flow through porous through homogenous porous media with mean capillary pressure. The solution is derived by the media; application of the Sumudu transform and the Fourier sine transform. The results are received in Capillary pressure; compact and graceful forms in terms of the generalized Mittag-Leffler function, which are suitable Generalized fractional for numerical computation. The mathematical formulation leads to generalized fractional derivative derivative; which has been solved by using a numerical technique by employing the iterative process with the Sumudu transform; help of appropriate boundary conditions. This problem has great importance in petroleum tech- Fourier sine transform; nology. Mittag-Leffler function ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction used in many areas of applied science and engineering. The oil– water movement in a porous medium is an important problem A porous medium is a material containing pores (voids). Voids of petroleum technology and water hydrology (Scheidegger, are usually filled with a fluid as liquid gas. A porous medium is 1966). Here we consider the injection of water into an oil for- most often characterized by its porosity. The skeletal portion mation in porous medium providing a two phase liquid–liquid of the material is often called the matrix or frame. Other prop- flow problem. Such a problem is generally encountered in sec- erties of the medium such as permeability, electrical conduc- ondary recovery process. A number of research workers have tivity and tensile strength can also be consequent for the also studied phenomenon of flow of two incompatible liquids respective properties of its constituents (solid matrix and fluid) through homogenous porous media with mean capillary pres- and the media porosity and pore structure, but these are gen- sure by using different mathematical resources (Bravo and erally complex. For a poroelastic medium the concept of por- Araujo, 2008; Brooks and Corey, 1964; Corey, 1954; osity is usually uncomplicated. This concept of porous media is Scheidegger, 1960; Scheidegger and Johnson, 1961). The frac- tional calculus has gained importance and popularity during the recent years or so, mainly due to its demonstrated applica- * Corresponding author. tions in science and engineering. For example, these equations E-mail addresses: [email protected] (A. Choudhary), are increasingly used to model problems in fluid flow, the- [email protected] (D. Kumar), jagdevsinghrathore@gmail. com (J. Singh). ology, diffusion, relaxation, oscillation, anomalous diffusion, reaction–diffusion, turbulence, diffusive transport akin to Peer review under responsibility of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2015.01.002 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 60 A. Choudhary et al. diffusion, electric networks, polymer physics, chemical physics, The fractional derivative of order a > 0 is presented electrochemistry of corrosion, relaxation processes in complex (Caputo, 1967) in the form: systems, propagation of seismic waves, dynamical processes in Z 1 x fðmÞ s self-similar and porous structures and many other physical c a ð Þ 0DxfðxÞ¼ amþ1 ds; m 1 < a < m processes (Hilfer, 2000; Srivastava et al., 2012; Moustafa and Cðm aÞ 0 ðx sÞ Salem, 2006; Podlubny, 1999; He, 1998; Chaurasia and dmfðxÞ ¼ ; if a ¼ m; m 2 N ð9Þ Singh, 2010). Many authors have proposed various methods dxm to handle linear and non-linear fractional differential equa- dmfðxÞ th where m is the m derivative of order m of the function fðxÞ tions which are of great importance in scientific and techno- dx logical fields. Among these are differential transform method with respect to x. The Sumudu transform of this derivative is (He, 1998; Atangana and Alabaraoye, 2013; Atangana and given (Chaurasia and Singh, 2010) as: Kilicman, 2013), homotopy perturbation method (Liu et al., Xm1 c a a aþk ðkÞ 6 2014), and variational iteration method (He and Wu, 2007). S½0DxfðxÞ; s¼u fðsÞ u f ð0þÞ; m 1 < a m: In this article, we study a fractional partial differential k¼0 equation associated with the generalized fractional derivative ð10Þ which is governed by the flow of immiscible phases in a A generalization of the Caputo fractional derivative opera- homogenous porous medium with initial and boundary condi- tor Eq. (9) is given (Hilfer, 2000), by introducing a right-sided tions. The solution of the fractional model is obtained by using fractional derivative operator of two parameters of order Sumudu and Sine transforms. 0 < a < 1 and 0 6 b 6 1 as:  d 2. Preliminary results Da;bfðxÞ¼Ibð1aÞ Ið1bÞð1aÞfðxÞ : ð11Þ 0 aþ aþ dx aþ The Sumudu transform of a function fðtÞ, determined for all If we put b ¼ 1, Eq. (11) reduces the Caputo fractional derivative operator assigned from Eq. (9). real numbers t P 0, is the function FsðuÞ, defined by Watugala (1993), Weerakoon (1994), Asiru (2001), and Sumudu transform formula for this operator is given by Belgacem and Karaballi (2005). Hilfer (2000), Belgacem et al. (2003), we hold: Z 1 S½ Da;bfðxÞ;s¼uafðsÞubða1Þþ1Ið1bÞð1aÞfð0þÞ; 0 < a 6 1; SffðtÞg ¼ FðuÞ¼GðuÞ¼ ð1=uÞet=ufðtÞdt: ð1Þ 0 x 0þ 0 ð12Þ We will also use the following outcome hold by Chaurasia where the initial value term and Singh (2011) as: ð1bÞð1aÞ I0þ fð0þÞ; ð13Þ 1 c1 b d c1 d b S ½u ð1 xu Þ ¼t Eb;cðxt Þ: ð2Þ involves the Riemann–Liouville fractional integral operator of The Fourier sine transform is defined by Debnath (1995). order ð1 bÞð1 aÞ evaluated in the limit as x ! 0þ. For Z more details and properties of this operator see Tomovski 2 1 Fðs; tÞ¼pffiffiffi fðx; tÞ sin sx dx: ð3Þ et al. (2010). p 0 The simplest Wright function is defined (Erde´lyi et al., 1981; Srivastava et al., 2012) as: The error function of x is defined by Rainville (1960) Z X1 1 zk 2 x /ða; b; zÞ¼ ; where a; b; z 2 C; ð14Þ erf x exp z2 dz 4 C ak b k! ð Þ¼ ð Þ ð Þ k¼0 ð þ Þ p 0 and the general Wright function is defined as: and the complimentary error function of x is defined as: Z "# Q x X1 p k ðai; aiÞð1;pÞ Cða þ a kÞ z 2 2 Qi¼1 i i erfcðxÞ¼ exp ðz Þ dz ð5Þ pwqðzÞ¼pwq z ¼ q ; ð15Þ ðbj; b Þ C b b k k! p 0 j ð1;qÞ k¼0 j¼1 ð j þ j Þ A generalization of the Mittag-Leffler function by Mittag- where z; a ; b 2 C and a ; b 2 Rði ¼ 1; 2; ...; p and Leffler (1903, 1905) i j i j j ¼ 1; 2; ...; qÞ then Eq. (15) reduces to familiar generalized X1 zn hyper-geometric function as (Thomas and George, 2006) EaðzÞ¼ ; ða 2 C; RðaÞ > 0Þð6Þ Cðna þ 1Þ X1 n¼0 ða Þ ...ða Þ zk F ða ; ...; a ; b ; ...; b ; zÞ¼ 1 k p k : ð16Þ p q 1 p 1 q b ... b k! was introduced (Wiman, 1905) in the general form k¼0 ð 1Þk ð qÞk X1 zn The generalized Navier–Stokes equations are given as E ðzÞ¼ ; ða; b 2 C; RðaÞ > 0Þð7Þ a;b C na b (Moustafa and Salem, 2006) n¼0 ð þ Þ X1 1 zk also derived (Shukla and Prajapati, 2007) in the following Wða; b; zÞ¼ ; where a; b; z 2 C: ð17Þ C ak b k! integral: k¼0 ð þ Þ Z 1 k ab st b1 d a k!s The relationship between the Wright function and the com- e t Ea;bðxt Þdt ¼ : ð8Þ k a kþ1 0 dz ðs xÞ plementary Error function is given as A fractional model of fluid flow through porous media 61   1 z @ K @P a;b o o 6 6 W; 1; z ¼ erfc : ð18Þ wDt So ¼ K ; where ð0 < a 1; 0 < b 1Þ: 2 2 @x fo @x ð29Þ

3. Mathematical model of the problem Eliminating @Pw from Eq. (28), we get: @x  a;b @ Kw @Po @Pc The seepage velocity ðUwÞ of water and oil ðfwÞ is assumed as wDt Sw ¼ K ; (Scheidegger, 1960) @x fw @x @x where ð0 < a 6 1; 0 < b 6 1Þ: ð30Þ K @P U ¼ w K w ð19Þ w f @x From Eqs. (29), (30) and (23) we get, w  @ Kw Ko @Po Kw @Pc Ko @Po K þ K ¼ 0: ð31Þ Uo ¼ K ð20Þ @x fw fo @x fw @x fo @x Now integrating Eq. (31), we get: and equation of continuity  @S @U Kw Ko @Po Kw @Pc w w þ w ¼ 0 ð21Þ K þ K ¼C; ð32Þ @t @x fw fo @x fw @x where C is the constant of integration, whose value can be @S @U w o þ o ¼ 0 ð22Þ calculated. @t @x K @P @P C K w c here K is considered as the permeability of the consistent medi- o ¼ þ fw @x @x Kw Ko Kw Ko um, Kw and Ko are the relative permeability of water and oil, K þ K þ fw fo fw fo which are the functions of the saturation of water ðSwÞ and oil ðSoÞ respectively, Pw and Po define the pressure of water @Pc @Po C @x and oil, aspect fw and fo are the kinematics viscosities of water ¼ þ ; ð33Þ @x K Kw 1 þ Ko fw 1 þ Ko fw and oil, while w is the medium of porosity and from the defini- fw fo Kw fo Kw tion of phase saturation (Scheidegger, 1960), it is apparent @Po that: Substituting the value of @x in Eq. (30) from Eq. (33),we got the conclusion that: Sw þ So ¼ 1 ð23Þ 8 0 1 9 < = @ K C @Pc K @P The capillary pressure ðPcÞ is defined as the pressure discon- Da;bS w K@ @x A w K c w t w ¼ : þ ; tinuity of the flowing phases across their common interface @x fw Kw Ko fw Ko fw fw @x K f 1þ f K 1þ f K which may be codified as: w o w o w 2 3 Pc ¼ Po Pw: ð24Þ @ Ko K @Pc C a;b 4fo @x 5 Relation between phase saturation and relative perme- wDt Sw þ þ ¼ 0 ð34Þ @x Ko fw Ko fw ability (Scheidegger and Johnson, 1961) is specified as: 1 þ f K 1 þ f K 9 o w o w > Kw ¼ Sw = Pressure of oil Po can be defined as:

Ko ¼ 1 Sw ð25Þ ;> Po þ Pw Po Pw 1 Po ¼ þ ¼ P þ Pc: ð35Þ Ko ¼ So 2 2 2 If the generalized fractional derivative model is used to pre- where P is constant, which is the mean pressure. sent the time derivative term, the equation of continuity is From Eqs. (32) and (35), we hold:  transformed into: K K K @P C ¼ w o c : ð36Þ @Uw 2 f f @x wDa;bS þ ¼ 0; where ð0 < a 6 1; 0 < b 6 1Þ; ð26Þ w o t w @x Substituting the value of C in Eq. (34), we hold: 2 3 @U a;b o Ko @Pc K Kw Ko @Pc wD So þ ¼ 0; where ð0 < a 6 1; 0 < b 6 1Þ: ð27Þ @ K ð Þ t @x a;b 4fo @x 2fw fo @x 5 wDt Sw þ þ ¼ 0 @x 1 þ Ko fw 1 þ Ko fw If we put a ¼ 1 and b ¼ 1 in Eqs. (26) and (27) reduced to fo Kw fo Kw Eqs. (21) and (22) respectively. 2 3 P K P 1 @ Ko K @ c þ w K @ c a;b 4 fo @x fw @x 5 4. Formulation of fractional partial differential equation wDt Sw þ ¼ 0 2 @x 1 þ Ko fw fo Kw

Putting the values of Uw and Uo in Eqs. (26) and (27) from Eqs.  @ K @P (19) and (20) respectively, we obtain the results: a;b 1 w c wDt Sw þ K ¼ 0  2 @x fw @x @ K @P wDa;bS ¼ w K w ; where ð0 < a 6 1; 0 < b 6 1Þ;  t w @x f @x w a;b 1 @ Kw @Pc @Sw wDt Sw þ K ¼ 0: ð37Þ ð28Þ 2 @x fw @Sw @x 62 A. Choudhary et al.

Kw @Pc which is the same solution as recently obtained (Prajapati Taking the K f @S ¼B then Eq. (37) is reduced as: w w et al., 2012). 2 1 @ Sw It can be written in terms of Wright function as: wDa;bS B ¼ 0 t w 2 @x2  a x S ðx; tÞ¼S W ; 1; pffiffiffiffiffiffiffi : ð45Þ @2S 1 w w0 2 lta w ¼ Da;bS ; ð38Þ x2 t w @ l If we set a ¼ 1 and make use of Eq. (12) and Eq. (45),we B arrive at the following result: where C ¼ 2P.  This is the partial differential equation of motion for satura- xffiffiffiffiffi Swðx; tÞ¼Sw0 erfc p : ð46Þ tion constraint conditions as: 9 2 lt Swðx; 0Þ¼0 => S ð0; TÞ¼S < 1 ð39Þ w w ;> 6. Conclusions limx!1Swðx; TÞ¼0; 0 < x < 1 In this paper, we have presented a fractional model of flow of 5. Analytical solution of the problem two incompatible liquids through homogenous porous media with mean capillary pressure. The solution has been developed in terms of Mittag-Leffler function by using the Sumudu Form Eq. (38), we include: transform and Fourier sine transform and their inverses after @2S deriving the related formulae for fractional integrals and Da;bS ðx; tÞ¼l w : ð40Þ t w @x2 derivatives. Operating Fourier sine transform Eq. (3) on Eq. (40),we Acknowledgments get: rffiffiffi Z 2 1 @2S The authors are very grateful to the anonymous referees for Da;bS s; t l w sin sx dx t wð Þ¼ 2 carefully reading the paper and for their constructive com- p 0 @x ments and suggestions which helped to improve the paper. Integrating by parts method, yields rffiffiffirffiffiffi Z 1 1 References a;b 2 @Sw 2 @Sw Dt Swðs; tÞ¼l sin sx sl cos sx dx p @x 0 p 0 @x rffiffiffirffiffiffi Asiru, M.A., 2001. Sumudu transform and the solution of integral 1 2 @Sw 2 equation of convolution type. Int. J. Math. Educ. Sci. Technol. 32, ¼ l sin sx sl ½S cos sx 1 w 0 906–910. p @x 0 p rffiffiffi Z Atangana, A., Alabaraoye, E., 2013. Solving system of fractional 1 2 2 partial differential equations arisen in the model of HIV infection s l Sw sin sx dx; p 0 of CD4+ cells and attractor one-dimensional Keller-Segel equa- tion. Adv. Differ. Equ. 2013 (article 94). and applying constraint conditions Eq. (40), we get: Atangana, A., Kilicman, A., 2013. The use of Sumudu transform for rffiffiffi rffiffiffi solving certain nonlinear fractional heat-like equations. Abstr. a;b 2 2 2 Appl. Anal. 2013, 12 (Article ID 737481). Dt Swðs; tÞ¼l ð0Þsl ð0 Sw0 Þs lSwðs; tÞ rpffiffiffi p Belgacem, F.B.M., Karaballi, A.A., 2005. Sumudu transform funda- 2 mental properties investigations and applications. Int. J. Appl. ¼sl S s2lS ðs; tÞ: ð41Þ Math. Stoch. Anal., 1–23 p w0 w Belgacem, F.B.M., Karaballi, A.A., Kalla, S.L., 2003. Analytical Now, using Eq. (12) and taking Sumudu transform for both investigations of the Sumudu transform and applications to sides in Eq. (41), we obtain: integral production equations. Math. Probl. Eng. 3, 103–118. rffiffiffi Bravo, M.C., Araujo, M., 2008. Analysis of the unconventional 2 1 behavior of oil relative permeability during depletion tests of gas- S~ ðs; uÞ¼sl S ½uaf1 ðs2lÞuag : ð42Þ w p w0 saturated heavy oils. Int. J. Multiph. Flow 34 (5), 447–460. Brooks, R.H., Corey, A.T., 1964. Hydraulic Properties of Porous Next, using Eq. (2) and taking inverse Sumudu transform Media. Hydrological Papers (Colorado State University) 3. for both sides in Eq. (42), we obtain: Caputo, M., 1967. Linear models of dissipation whose Q is almost rffiffiffi frequency independent-II. Geophys. J. Royal Astron. Soc. 13, 529– 2 a 2 a 539 (Reprinted. In: Fract. Calc. Appl. Anal., 11(1), 2008, 3–14). Swðs; tÞ¼sl Sw0 t Ea;aþ1ðs lt Þ: ð43Þ p Chaurasia, V.B.L., Singh, J., 2010. Application of Sumudu transform Finally, taking the inverse Fourier sine transform on above in schrodinger equation occurring in Quantum Mechanics. Appl. Eq. (43), we get: Math. Sci. 4 (57), 2843–2850. rffiffiffi "#rffiffiffi Z Chaurasia, V.B.L., Singh, J., 2011. Application of Sumudu transform 2 2 1 in fractional kinetic equations. Gen. Math. Notes 2 (1), 86–95. S ðx; tÞ¼l S ta fsE ðs2ltaÞg sin sx ds ; w p w0 p a;aþ1 Corey, A.T., 1954. The interrelation between gas and oil relative 0 permeabilities. Prod. Monthly 19 (1), 38–41. Z Debnath, L., 1995. Integral Transforms and Their Applications. CRC 1 2 a 2 a Press, New York, London, Tokyo. Swðx; tÞ¼ lSw0 t fsEa;aþ1ðs lt Þg sin sx ds ð44Þ p 0 A fractional model of fluid flow through porous media 63

Erde´lyi, A., Magnus, W., Oberhettinger, F., Tricomi, F., 1981. Higher Scheidegger, A.E., 1960. The Physics Flow through Porous Media, Transcendental Functions. Krieger, Melbourne, 3. 216. University of Toronto Press (pp. 229–231). He, J.H., 1998. Approximate analytical solution for seepage flow with Scheidegger, A.E., 1966. Flow through Porous Media, as in Applied fractional derivatives in porous media. Comput. Meth. Appl. Mechanics Surveys by Abramson. Spartan Books, Washington. Mech. Eng. 167, 57–68. Scheidegger, A.E., Johnson, E.F., 1961. The statistical behaviour of He, J.H., Wu, X.H., 2007. Variational iteration method: new devel- instabilities in displacement process in porous media. Can. J. Phys. opment and applications. Comput. Math. Appl. 54, 881–894. 39 (2), 326. Hilfer, R., 2000. Applications of Fractional Calculus in Physics. World Shukla, A.K., Prajapati, J.C., 2007. On a generalization of Mittag- Scientific Publishing Company, Singapore, New Jersey, Hong Leffler function and its properties. J. Math. Anal. Appl. 336, 797– Kong (pp. 87–130). 811. Liu, H.Y., He, J.H., Li, Z.B., 2014. Fractional calculus for nanoscale Srivastava, H.M., Parmar, R.K., Chopra, P., 2012. A class of extended flow and heat transfer. Int. J. Numer. Meth. Heat Fluid Flow 24 fractional derivative operators and associated generating relations (6), 1227–1250. involving hypergeometric functions. Axioms 1 (3), 238–258. Mittag-Leffler, G.M., 1903. Sur la nouvelle function Ea (x). CR Acad. Thomas, C., George, C., 2006. The Fox–Wright functions and Sci. Paris 137, 554–558 (ser. II). Laguerre multiplier sequences. J. Math. Anal. Appl. 314, 109–125. Mittag-Leffler, G.M., 1905. Sur la representatin analytique d’une Tomovski, Z., Hilfer, R., Srivastava, H.M., 2010. Fractional and branche uniforme d’une function monogene. Acta Math. 29, 101– operational calculus with generalized fractional derivative opera- 181. tors and Mittag-Leffler type functions. Integral Transforms Spec. Moustafa, El.-Shahed, Salem, A., 2006. Decay of vortex velocity and Funct. 21 (11), 797–814. diffusion of the temperature for fractional viscoelastic fluid through Watugala, G.K., 1993. Sumudu transform a new integral transform to porous medium. Int. J. Commun. Heat Mass Transfer 33, 240–248. solve differential equations and control engineering problems. Int. Podlubny, I., 1999. Fractional Differential Equations. Academic Press, J. Math. Educ. Sci. Technol. 24, 35–43. New York. Weerakoon, S., 1994. Applications of Sumudu transform to partial Prajapati, J.C., Patel, A.D., Pathak, K.N., Shukla, A.K., 2012. differential equations. Int. J. Math. Educ. Sci. Technol. 25, 277– Fractional calculus approach in the study of instability phe- 283. nomenon in fluid dynamics. Palestine J. Math. 1 (2), 95–103. Wiman, A., 1905. Uber den fundamentalsatz in der theorie der Rainville, E.D., 1960. Special Functions. The Macmillan Company, functionen Ea (x). Acta Math. 29, 191–201. New York. Journal of the Association of Arab Universities for Basic and Applied Sciences (2016) 21,64–67

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ORIGINAL ARTICLE New interaction solutions to the combined KdV–mKdV equation from CTE method

Hengchun Hu a,b,*, Meiying Tan a, Xiao Hu a a College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China b Department of Mathematics, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

Received 11 September 2015; revised 25 November 2015; accepted 13 January 2016 Available online 19 February 2016

KEYWORDS Abstract The consistent tanh expansion (CTE) method is developed for the combined KdV– Consistent tanh expansion; mKdV equation. The combined KdV–mKdV equation is proved to be CTE solvable. New exact Soliton–cnoidal wave; interaction solutions such as soliton–cnoidal wave solutions, soliton–periodic wave solutions for Soliton–periodic solution the combined KdV–mKdV equation are given out analytically and graphically. Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction types of nonlinear excitations such as the soliton-resonant solution, soliton–cnoidal waves and soliton–perodic waves. As well known, how to find abundant more exact solutions for Many new interaction solutions for nonlinear systems, for nonlinear systems, especially interaction solutions, is one of instance, the Boussinesq equation, dispersive water wave equa- the most important aspects in the soliton theory. Many inte- tion, Boussinesq-Burgers equation, break soliton equation, grable properties, such as the multi-soliton solutions, Darboux nonlinear Schro¨ dinger equation and modified Kadomtsev- transformation, symmetry reduction, Hirota bilinear form, Petviashvili equation, are discussed in detail (Alam and homogeneous balance method, etc, are studied extensively Akbar, 2015; Bekir, 2009; Hu et al., 2012; Lou et al., 2014; after efforts of mathematicians and physicists. But for other Ren, 2015). integrable systems, it is still very difficult to find the interaction In this paper, we focus on the combined KdV–mKdV equa- solutions among different types of excitations because there tion, which is also known as the Gardner equation are no universal formulae to construct all the possible interac- u þ 2auu 3u2u þ u ¼ 0; ð1Þ tion wave solutions. However, according to the results of the t x x xxx symmetry reduction with nonlocal symmetries, Lou proposed where a is a constant. Eq. (1) is widely used in various fields of the consistent tanh expansion method (CTE) recently (Lou, physics, such as solid-state physics, plasma physics, fluid phy- 2015), which is a more generalized but a much simpler method sics and quantum field theory (Fu et al., 2004; Miura, 1997; Xu to find new interaction solutions between a soliton and other et al., 2003). The solitary solutions, traveling wave solutions, quasi-periodic solutions and symmetries for the combined * Corresponding author at: College of Science, University of KdV–mKdV Eq. (1) have been studied by means of the Shanghai for Science and Technology, Shanghai 200093, PR China. extended mapping method, extended tanh expansion method E-mail address: [email protected] (H. Hu). and new Riccati equation expansion method (Bekir, 2009; Peer review under responsibility of University of Bahrain. Huang and Zhang, 2006 Sirendaoerji, 2006; Zhao et al., http://dx.doi.org/10.1016/j.jaubas.2016.01.005 1815-3852 Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). New interaction solutions to the combined KdV–mKdV equation 65  2006). The generalization form in (2+1)-dimensions of the w2 w xx 2w2 xxx 12w2 6S 6C 2a2 2 þ x ð x Þ combined KdV–mKdV equation has been discussed in Dai wx wx and Wang (2014), Geng and Cao (2001), Krishnan and Peng ð12w2 6S 6C 2a2Þ (2006), Zhang and Lin (1995), and Zhou and Ma (2000). x xx wxx The paper is organized as follows. In Section 2, the CTE ð12w2 6S 6C 2a2Þ ¼ 0; ð8Þ w x x method will be defined briefly. New different interaction solu- x tions including soliton–cnoidal waves, soliton–periodic waves where solutions are obtained in Section 3. The detailed structures 2 wxxx 3 w wt of new interaction solutions among nonlinear excitations are S ¼ xx ; C ¼ ; w 2 w2 w also given out graphically. The last section is devoted to sum- x x x mary and discussion. are Mo¨ bius invariants. It is easy to see that the Eqs. (7) and (8) are satisfied automatically because of (6) and we know that the 2. Consistent tanh expansion for the combined KdV–mKdV w-consistent condition is just Eq. (6). From Eqs. (7) and (8),we equation can also have Eq. (6). Here we just consider the simplest Eq. (6) about w as the consistent condition to avoid the compli- cated calculation. In other words, if w is a solution of (6), then For a given nonlinear evolution system, ffiffiffi p ffiffiffi ; ; ; ; ; ...; ; ; ; ...; ; 2awx 3 2rwxx p Pðx t uÞ¼0 x ¼ xðx1 x2 xnÞ u ¼ uðu1 u2 unÞ u ¼ þ 2rw tanhðwÞ; ð9Þ 6w x ð2Þ x is also a solution of the combined KdV–mKdV Eq. (1). So the we aim to look for the possible truncated expansion solution expression (9) can be regarded as a nonauto-Ba¨ cklund trans- Xn formation of (1). Once the solution of (6) is known, then the u ¼ u ðx; tÞtanh jðwÞ; ð3Þ j solution of (1) will be obtained directly from (9). Many more j¼0 interesting concrete interaction solutions will be studied in where wðx; tÞ is a function to be determined, n should be deter- detail in the next section. mined from the leading order analysis of Eq. (2) and all the ; expansion coefficients ujðx tÞ will be determined by vanishing 3. New interaction solutions for combined KdV–mKdV equation the coefficients of different powers of tanhðwÞ after substitut- ing Eq. (3) into Eq. (2). If the system of u and w is consistent, j It has been pointed out that if the solution of w-equation (6) is we call the nonlinear system (2) is CTE solvable. In order to 2 known, one can obtain the explicit solutions of Eq. (1) from balance the nonlinear term u ux and the dispersive term uxxx, (9). However, it is difficult to find the general solution of (6) we have 3n þ 1 ¼ n þ 3 and it is easy to find n ¼ 1, which is because of its complexity. In order to obtain the interaction very similar to the leading order analysis of the Painleve´test solutions between solitons and cnoidal waves of Eq. (1),we for the nonlinear differential equation. We can seek the follow- consider the function w in the form ing truncated tanh expansion w ¼ k1x þ x1t þ WðXÞ; X ¼ k2x þ x2t: ð10Þ u ¼ u0 þ u1 tanhðwÞ: ð4Þ Substituting (10) into Eq. (6), we can find that W ðXÞ Substituting Eq. (4) into Eq. (1) yields 1 satisfies 3u w ðu2 2w2 Þtanh4ðwÞþð6u w w 2au2w 3u2u 1 x 1 x 1 x xx 1 x 1 1x C C C 2 4 3 2 2 1 0 ; 2 2 3 2 W1X ¼ 4W1 þ C3W1 þ W1 þ W1 þ ð11Þ þ 6u0wxu1 þ 6u1xwxÞtanh ðwÞþ½2au1u1x 3u1ðu0x þ u1wxÞ 3 3 3 3 2 with u1wt þ 8u1wx 3u1xwxx 2au0u1wx þ 3u0u1wx u1wxxx

2 WðXÞ ¼ W1ðXÞ; 6u0u1u1x 3u1xxwxtanh ðwÞþ½u1t þ u1xxx þ 2au0u1x X 6x 9C k k2 2a2k 72k k2 2 2 2 þ 3 1 2 þ 2 2 1 ; 6u1wxwxx þ 2au1wx þ 2au1u0x 6u0u1u0x 6u0u1wx C2 ¼ 3 k2 2 2 3u u 6u w tanh w u u 3u w u w 2 2 2 3 0 1x 1x x ð Þþ 0t þ 0xxx þ 1xx x þ 1 xxx 9C k k þ 3k x þ 9k x þ 4a k k 96k k C 3 1 2 2 1 1 2 1 2 1 2 ; 2 3 2 1 ¼ 4 þ 2au0u0x þ u1wt þ 2au0u1wx 3u0u1wx 2u1wx 3u0u0x k2 2 3 2 2 : k1ð2a k1k2 36k k2 þ 3k2x1 þ 3C3k k þ 3k1x2Þ þ 3u1xwxx ¼ 0 1 1 2 ; C0 ¼ 5 k2 Then setting the coefficients of different powers of tanh jðwÞ to zero and solving the undetermined functions u0; u1; w,we and C3 is an arbitrary constant. It is known that the general have solutions of Eq. (11) can be expressed in terms of Jacobi elliptic pffiffiffi functions. The concrete examples of the soliton–cnoidal wave r pffiffiffi 2awx 3 2 wxx 2 interaction solutions will be discussed in the following paper. u0 ¼ ; u1 ¼ 2rwx; r ¼ 1; ð5Þ 6wx Firstly, we assume the function w in the form and w ¼ k0x þ x0t þ c1Efðsnðk1x þ x1t; mÞ; mÞ; ð12Þ 12w2 6S 6C 2a2 ¼ 0; ð6Þ x where Ef is the first incomplete elliptic integral and sn is the w 2 xx 2 2 ; usual Jacobi elliptic sine function. Substituting (12) into Eq. ð12wx 6S 6CÞx þ ð12wx 6S 6C 2a Þ¼0 ð7Þ wx (6) and setting the coefficients of different powers of Jacobi 66 H. Hu et al. elliptic functions into zero, we will have two constant solu- tions. The first case is

2 2 x k0ð48k0k1 2a k1 3 1Þ k0 x0 ¼ ; c1 ¼ ; ð13Þ 3k1 k1 with k0; k1; m; x1; a being arbitrary constants. The second case is

k ð6k2 þ 18k2c2 a2Þ 1 x ¼ 0 0 1 1 ; x ¼ 2c2k3 þ 6k2k a2k ; 0 3 1 1 1 0 1 3 1 ð14Þ where c1; k0; k1; m; a are five arbitrary constants. Then substi- tuting (12) with Eqs. (13) or (14) into Eq. (9) respectively, we will obtain different types of the soliton–cnoidal wave interac- tion solutions for the combined KdV–mKdV Eq. (1). We omit the complicated expression of the soliton–cnoidal wave inter- action solutions and only the structure is shown in Fig.1 by selecting the arbitrary constants as a ¼1; x1 ¼3; k0 ¼0:5; k1 ¼ 2; m ¼ 0:98; r ¼ 1: ð15Þ Fig. 2 Second type of soliton–cnoidal wave interaction solution When the arbitrary constants are fixed as of u given by (9), (12), (14) and (16). c1 ¼ 0:5; a ¼ 1; k0 ¼1; k1 ¼ 2; m ¼ 0:98; r ¼ 1; ð16Þ substituting (17) into the consistent condition (6), we can arrive at the relations of constants we can obtain new soliton–cnoidal wave interaction solution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for combined KdV–mKdV Eq. (1) given by (9), (12), (14) 2 2 k3ðc2 n þ 1Þ nð1 nÞðc2 n þ 1Þ and (16) and the detailed structure is given in Fig. 2. k2 ¼ ; m ¼ ; ð18Þ c 1 n Secondly, in order to find more interesting soliton–cnoidal 2 1 À x 2 2 4 2 2 2 wave interaction solution, we also can restrict the function w as 2 ¼ 3 ½ð18n 12Þk3 þ a c2 ðn 1Þ½18k3ðn 1Þþa c2 3c2 x x ; ; ; ;  w ¼ k2x þ 2t þ c2Ep½snðk3x þ 3t mÞ n m ð17Þ 2nk2c3 þ 6ðn 1Þ3k2 þ 3 2 ; ð19Þ 3 1 n where the function Ep is the third incomplete elliptic integral hi 2nk3c2 k and m is the module of the Jacobi elliptic sine function. After x 3 2 3 2 2 2 2 2 ; 3 ¼ 2 ð12k3ð2n 1Þþa Þc2 18k3ðn 1Þ ð20Þ n 1 3c2

with k3; c2; n; a being arbitrary constants. Substituting Eqs. (17)–(20) into Eq. (9) and selecting the arbitrary constants as

c2 ¼ 0:5; n ¼ 0:5; k3 ¼ 2; a ¼ 1; r ¼ 1; ð21Þ

we will have another interesting soliton–cnoidal wave interac- tion solution of the combined KdV–mKdV equation which is displayed in Fig. 3. Lastly, it is known that the single soliton and periodic wave solutions expressed by hyperbolic functions and Jacobi elliptic functions have been studied in Zhao et al. (2006), Sirendaoerji (2006),Huang and Zhang (2006), Zhu (2014) by means of the extended tanh expansion method and Jacobi elliptic function expansion. It is clear that trivial solution of the consistent con- dition (6)  1 w ¼ kx þ 2k3 a3k t; 3

leads to the single soliton for the combined KdV–mKdV equa- tion. So we can obtain not only the usual soliton solution but also many new soliton–cnoidal wave interaction solutions in Fig. 1 First type of soliton–cnoidal wave interaction solution of Figs. 1–3 for nonlinear integrable systems from the CTE u given by (9), (12), (13) and (15). method. New interaction solutions to the combined KdV–mKdV equation 67

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University of Bahrain Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com

ORIGINAL ARTICLE A simple harmonic balance method for solving strongly nonlinear oscillators

Md. Abdur Razzak

Dept. of Mathematics, Rajshahi University of Engineering and Technology (RUET), Kazla, Rajshahi-6204, Bangladesh

Received 27 June 2015; revised 27 August 2015; accepted 11 October 2015 Available online 28 November 2015

KEYWORDS Abstract In this paper, a simple harmonic balance method (HBM) is proposed to obtain higher- Nonlinear oscillation; order approximate periodic solutions of strongly nonlinear oscillator systems having a rational and Harmonic balance method; an irrational force. With the proposed procedure, the approximate frequencies and the correspond- Truncation ing periodic solutions can be easily determined. It gives high accuracy for both small and large amplitudes of oscillations and better result than those obtained by other existing results. The main advantage of the present method is that its simplicity and the second-order approximate solutions almost coincide with the corresponding numerical solutions (considered to be exact). The method is illustrated by examples. The present method is very effective and convenient method for solving strongly nonlinear oscillator systems arising in nonlinear science and engineering. Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction et al., 2006; Lim et al., 2005; Alam et al., 2007; Hosen et al., 2012) is another technique for solving strongly nonlinear equa- Nonlinear oscillation problems are essential tool in physical tions. When a HBM is applied to the nonlinear equations for science, mechanical structures, nonlinear circuits, chemical higher-order approximation, then a set of difficult nonlinear oscillation and other engineering research. Nonlinear vibra- complex equations appear and it is very difficult to analytically tions of oscillation systems are modeled by nonlinear differen- solve these complex equations. In a recent article, Hosen et al. tial equations. It is very difficult to obtain periodic solutions of (2012) solved such nonlinear algebraic equations easily by such nonlinear equations. There are several methods used to using a truncation principle. Recently, many authors (Khan solve nonlinear differential equations. Among one of the et al., 2011, 2012a,b, 2013a; Khan and Mirzabeigy, 2014; widely used is perturbation method (Marion, 1970; Krylov Saha and Patra, 2013; Yazdi et al., 2010; Yildirim et al., and Bogoliubov, 1947; Bogoliubov and Mitropolskii, 1961; 2011a,b, 2012; Khan and Akbarzade, 2012; Akbarzade and Nayfeh and Mook, 1979) whereby the nonlinear response is Khan, 2012; Akbarzade and Khan, 2013) have studied small. On the other hand, there are many methods (Amore strongly nonlinear oscillators. Khan et al. (2012a) used a cou- and Aranda, 2005; Cheung et al., 1991; He, 2002) used to solve pling method combining homotopy and variational approach. strongly nonlinear equations. The harmonic balance method Other authors (Nayfeh and Mook, 1979; Mickens, 2001; Hu (HBM) (Belendez et al., 2007; Mickens, 1996, 1984; Wu and Tang, 2006; Lim and Wu, 2003) used HBM to solve some strongly nonlinear oscillators. But it is a very laborious proce- E-mail address: [email protected] dure to obtain higher-order approximations using those meth- ods (Nayfeh and Mook, 1979; Mickens, 2001; Hu and Tang, Peer review under responsibility of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2015.10.002 1815-3852 Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Harmonic balance method for solving strongly nonlinear oscillators 69

2006; Lim and Wu, 2003). Fesanghary et al. (2009) obtained a For the first-order approximation (putting n = 0 and new analytical approximation by variational iterative method u3 = u5 = = 0 in Eq. (3)), Eq. (3) becomes (VIM); but the solution contains many harmonic terms. Many x ðtÞ¼A cos u: ð4Þ analytical techniques (Hosen et al., 2012; Mickens, 1996, 2001; 0 Lim and Wu, 2003; Tiwari et al., 2005; Ozis and Yildirim, Eq. (4) also satisfied Eq. (2). 2007; Ghadimi and Kaliji, 2013; Ganji et al., 2009; Zhao, In this paper, Eq. (1) can be re-written as 2009; Akbarzade and Farshidianfar, 2014; Khan et al., x€ þ x2x þ fðxÞ 2013a,b) have been used to solve strongly nonlinear oscillator 0 ¼ 0: ð5Þ 1 þ x2 systems having a rational force (such as Duffing-harmonic 0 € x3 Using Eqs. (3) and (4), we have the left-side the following oscillator: x þ 1þx2 ¼ 0 etc.) and irrational force (mass attached to a stretched wire: x€ þ x pffiffiffiffiffiffiffiffix ¼ 0 etc.). Hosen et al. (2012) Fourier series expansions: 1þx2 € x2 solved the Duffing-harmonic oscillator by expanding the term x þ 0x þ fðxÞ ¼ c1 cos u þ c3 cos 3u þ c5 cos 5u þ;; ð6Þ 3 2 x € 3 5 ð1 þ x0Þ 1þx2 into a polynomial form x þ x x þ¼0. But this method (Hosen et al., 2012) is valid for small amplitude of where oscillations and it is invalid to investigate the nonlinear oscil- Z p  2 € x2 € kffiffiffiffiffiffiffiffix 4 x þ 0x þ fðxÞ lator x þ x p ¼ 0. On the contrary, other authors c2n1 ¼ cosð2n 1Þu du; n ¼ 1;2;3;;: 1þx2 p 2 0 1 þ x0 (Fesanghary et al., 2009; Khan et al., 2013a,b) have used dif- ð7Þ ferent analytical techniques to solve these nonlinear oscillators 3 Substituting Eq. (6) for Eq. (5) and then equating the coef- € x € pkffiffiffiffiffiffiffiffix x þ x2 ¼ 0; x þ x ¼ 0 etc. without expanding. But 1þ 1þx2 ficients of the terms cos u and cos 3u, cos 5u,..., we get a set their solution procedure for determining higher-order approx- of nonlinear algebraic equations whose solutions provide the imations of these nonlinear oscillators is not easy or straight- unknown coefficients u3, u5, together with the frequency, x. forward and the results (obtained by second order approximation) are not more accurate compared with numer- 3. Examples ical results. The purpose of this paper is to apply a simple factor on the 3.1. Example 1 strongly nonlinear oscillator systems having a rational and an irrational force and to obtain higher-order approximate fre- Let us consider a one-dimensional, nonlinear Duffing- quencies and the corresponding periodic solutions by easily harmonic oscillator of the form (Mickens, 2001) solving the sets of algebraic equations with complex nonlinear- x3 ities. The trial solution (concern of this paper) is the same as x€ þ ¼ 0; ð8Þ that of Hosen et al. (2012). But the solution procedure is differ- 1 þ x2 ent from that of Hosen et al. (2012). To verify the accuracy of with initial conditions the present method, the two complicated nonlinear oscillators xð0Þ¼A; x_ð0Þ¼0: ð9Þ € x3 € pkffiffiffiffiffiffiffiffix (x þ x2 ¼ 0; x þ x ¼ 0) are chosen as examples. The 1þ 1þx2 Eq. (1) is an example of a conservative nonlinear oscillatory method provides better result for both small and large ampli- system having a rational form for the non-dimensional restor- tudes of oscillations. The significance of this present method is ing force. its simplicity, which not only provides a few harmonic terms, Mickens (2001) rearranged Eq. (8) as but also gives more accurate measurement than any other exit- 2 € 3 : ing solutions. ð1 þ x Þx þ x ¼ 0 ð10Þ Applying the lowest order harmonic balance method to Eq. 2. The methods (10), Mickens (2001) obtained an approximate solution of this oscillator. For the higher-order approximation solutions, a set Let us consider the following general strongly nonlinear oscil- of complicated algebraic equations are involved and it is very lator systems having a rational or an irrational force: difficult to analytically solve. On the other hand, Fesanghary x€ þ x2x þ fðxÞ¼0; ð1Þ et al. (2009) obtained higher-order solutions (containing up 0 to ninth harmonic terms) from Eq. (10). To overcome these with initial conditions problems, approximation solutions (containing up to third xð0Þ¼A; x_ð0Þ¼0; ð2Þ harmonic terms) have been obtained by applying an easy approach to Eq. (10) which is based on HBM. In this article, where over dot denotes the derivatives with respect to t, A nonlinear algebraic equations are solved by truncating higher denotes the maximum amplitude, f(x) is a nonlinear order terms (followed partially by the principle rule presented restoring-force function such that f x f x and x P 0: ð Þ¼ð Þ 0 in Hosen et al. (2012)). The approximate periodic solution of Eq. (1) is taken in the Consider the second-order approximate periodic solution form similar to that of Hosen et al. (2012) of Eq. (8) is of the form xnðtÞ¼Aðð1 u3 u5 Þcos u þ u3 cos 3u þ u5 cos 5u þÞ; x1ðtÞ¼Aðð1 u3Þ cos u þ u3 cos 3uÞ: ð11Þ n ¼ 0; 1; 2; ; ð3Þ Therefore, the first-order approximation becomes where u = xt, x is an unknown angular frequency and x0ðtÞ¼A cos u: ð12Þ u3; u5; are constants which are to be further determined. 70 M.A. Razzak

2 x2 x2 x2 2 3 ; In this paper, by dividing Eq. (10) by the factor ð1 þ x0Þ b0 A þðb1 þ b2Þu3 þðb3 þ b4Þu3 þ Oðu3Þ¼0 ð16Þ and then substituting Eqs. (11) and (12) for Eq. (10) and also x x2 2 : using Eqs. (6) and (7), finally, the approximate frequency d0 þðd1 þ d2Þu3 þ Oðu3Þ¼0 ð17Þ and the approximate periodic solution can be found. It is 2 At first take u3 = 0, Eq. (16) becomes noted that dividing by the factor (ð1 þ x0Þ), a set of simple algebraic equations appears which contains lower order terms 2 b0 x A ¼ 0: ð18Þ and these lower order terms make the solution rapidly con- verge. On the other hand, the results obtained in this paper Solving Eq. (18), we obtain give more accuracy than other existing results. rffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 2 We re-write Eq. (8) in the form x x 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1ðAÞ¼ ¼ 1 2 þ ð19Þ A A A2 A2 2 € 3 1 þ ð1 þ x Þx þ x : 2 ¼ 0 ð13Þ 1 þ x0 The first approximate frequency is given in Eq. (19) which was also obtained by Lim and Wu (2003). Using Eqs. (11) and (12) in Eq. (13), we have the following It is obvious that the frequency as well as solution Eq. (11) Fourier series expansions: gives better result when Eqs. (16) and (17) are truncated 2 € 3 ð1 þ x Þx þ x u u ; (Hosen et al. (2012)). Using the truncation rule of Hosen 2 ¼ c1 cos þ c3 cos 3 þ ð14Þ 1 þ x0 et al. (2012) in the Eqs. (16) and (17), we obtain the following results where Z  x2 x2 x2 2= ; p b0 A þðb1 þ b2Þu3 þðb3 þ b4Þu3 2 ¼ 0 ð20Þ 4 2 ð1 þ x2Þx€þ x3 c ¼ cosð2n 1Þu du; n ¼ 1;2;3;: 2n1 p 2 0 1 þ x 2 0 d0 þðd1 þ x d2Þu3 ¼ 0: ð21Þ ð15Þ Eliminating x from Eqs. (20) and (21), we obtain The first two terms have been obtained as = 2 : Ad0 þðb2d0 Ad1 b0d2Þu3 þðb2d1 b1d2 þb4d0 2Þu3 þ¼0 c b x2A b x2b u b x2b u2 O u3 ; 1 ¼ 0 þð 1 þ 2Þ 3 þð 3 þ 4Þ 3 þ ð 3Þ ð22Þ c ¼ d þðd þ x2d Þu þ Oðu2Þ; 3 0 1 2 3 3 Solving Eq. (22), we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where 2 "#pðAÞ q ðAÞ8Ad0qðAÞ 2 4 2 u3 ¼ ; ð23Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; 24 A A p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ A ; 2qðAÞ b0 ¼ A þ b1 ¼ 3 1 þ A A A2 A 2 8 A2 "#1 þ 1 þ pðAÞ¼2b2d0 2Ad1 2b0d2; qðAÞ¼2b1d2 b4d0 2b2d1: 16 A2 3A4 1 þ A2 b 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ¼ 3 þ þ Solving Eq. (20), we obtain the second approximate fre- A 2 16 1 þ A2 "#quency as 2 4 6 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 96 3A 3A A 1 þ 2A þ A 2= b3 ¼ 1 þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b0 b1u3 b3u3 2 5 2 xðAÞ¼x ðAÞ¼ ; ð24Þ A 2 8 16 1 þ A 2 2= "#A b2u3 þ b4u3 2 16 41A4 11A6 2ð19 þ 34A2 þ 15A4Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; where u is given by Eq. (23). b4 ¼ 5 38 þ 49A þ 3 A 4 8 1 þ A2 Therefore, the second-order approximate periodic solution of Eq. (8) is 8 2 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 ; d0 ¼ þ = 3 3 2 1 2 2 x x ; A A A A x1ðtÞ¼Aðð1 u3Þ cos t þ u3 cos 3 tÞ ð25Þ "#A ð1 þ A Þ 1 þ 24 A6 4 7A2 3A4 x 2 4 þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ ; where u3 and respectively, are given by Eqs. (23) and (24). d1 ¼ 5 4 5A A þ þ A 8 1 þ A2 In a similar way, a third-order approximate solution, "# x2ðtÞ¼Aðð1 u3 u5Þ cos xt þ u3 cos 3xt þ u5 cos 5xtÞð26Þ 16 11A6 4 7A2 3A4 2 4 þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ : d2 ¼ 5 4 þ 5A þ A A 16 1 þ A2 is found for which the related equations are obtained as x2 x2 x2 2 Here, we see that the above coefficients are not singular for b0 A þðb1 þ b2Þu3 þðb3 þ b4Þu3 2 2 A tending to zero. For example, þðb5 þ x b6Þu3u5=2 þðb7 þ x b8Þu5 ¼ 0; ð27Þ 2 2 2A pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 2 2 b0 ¼ A þ ¼ A d0 þðd1 þ x d2Þu3 þðd3 þ x d4Þu þðd5 þ x d6Þu5 ¼ 0; ð28Þ A A 1 þ A2 1 þ A2ð 1 þ A2 þ 1Þ 3 x2 x2 2 x2 : So, r0 þðr1 þ r2Þu3 þðr3 þ r4Þu3 þðr5 þ r6Þu5 ¼ 0 ð29Þ

lim b0 ¼ 0: where b0, b1, b2, b3, b4; d0, d1, d2 are already defined above and A!þ0 "# Substituting Eq. (14) for Eq. (13) and then equating the 192 A6 A8 4 9A2 6A4 A6 2 4 ð þ þ þ Þ ; coefficients of the terms cos u and cos 3u equal to zeros, b5 ¼ 4 þ 7A þ 3A þ þ = A7 8 32 2 1 2 respectively, we obtain ð1 þ A Þ Harmonic balance method for solving strongly nonlinear oscillators 71 " 6 8 64 59A A ðAd1 þ b0d2 b2d0ÞðAd5 þ b0d6 b8d0Þr0 2 2 4 u3 ¼ l þ l b6 ¼ 7 140 237A 105A 2 A 8 16 Ad0ðAr5 þ b0r6 b8r0Þ # 3 2 4 6 þðb1d2=Ad0 b4=A þÞl þ; ð31Þ ð140 þ 307A þ 206A þ 39A Þ ; þ = 2 1 2 ð1 þ A Þ 1 þðr1=r0 þ b0r2=ðAr0Þb2=AÞl u5 ¼ þ; "# ðb8=A r5=r0 b0r6=ðAr0ÞÞ 24 A4 4 5A2 A4 2 þ þ ; = ; 1 b7 ¼ 4 3A þ = b8 ¼2b7 3 l : 5 2 1 2 ¼ ð32Þ A 8 ð1 þ A Þ ðb2=A d1=d0 b0d2=ðAd0ÞÞ "# 96 7A6 3A6 4 11A2 10A4 3A6 2 4 þ þ þ ; 3.2. Example 2 d3 ¼ 7 4 þ 9A þ 6A þ 1=2 A 8 32 ð1 þ A2Þ " In dimensionless form, we consider a mass attached to the cen- ter of a stretched elastic wire has the equation of motion (Sun 32 101A6 41A6 d ¼ 76 155A2 94A4 þ et al. (2007)) 4 A7 8 32 # k € ffiffiffiffiffiffiffiffiffiffiffiffiffix ; 76 193A2 162A4 45A6 x þ x p ¼ 0 ð33Þ þ þ þ ; 1 þ x2 þ 1=2 1 A2 ð þ Þ where over dots denote differentiation with respect to time t "#and 0 < k 6 1. 24 A6 16 32A2 19A4 3A6 2 4 þ þ þ ; The initial conditions are d5 ¼ 16 þ 24A þ 9A þ = A7 2 2 1 2 ð1 þ A Þ xð0Þ¼A; x_ð0Þ¼0: ð34Þ d ¼2d =3; 6 5 Eq. (33) is an example of conservative nonlinear oscillatory "#system having an irrational elastic form for the restoring force. 2 16 20A2 5A4 2 4 þ þ ; Eq. (33) can be re-written as r0 ¼ 5 16 12A A þ 1=2 A ð1 þ A2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi "#1 þ x2ðx€ þ xÞkx ¼ 0 ð35Þ 24 16 36A2 25A4 5A6 2 4 6 þ þ þ ; r1 ¼ 16 þ 28A þ 13A þ A = In this paper, we re-write Eq. (35) in the following form A7 2 1 2 ð1 þ A Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x2ðx€ þ xÞkx " pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0: ð36Þ 2 32 2 4 6 8 1 þ x0 r2 ¼2r1=3; r3 ¼ 48 132A 123A 42A 3A A9 # Using Eqs. (11) and (12) in Eq. (36), we have the following 3A10 48 þ 156A2 þ 183A4 þ 90A6 þ 15A8 Fourier series expansions: ; ffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ 1=2 p 32 ð1 þ A2Þ 1 þ x2ðx€ þ xÞkx pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c1 cos u þ c3 cos 3u þ; ð37Þ 2 " 1 þ x0 128 2 4 6 8 r4 ¼ 9 304 þ 772A þ 667A þ 214A þ 15A where A Z p  # 2 3 4 2 ð1 þ x Þx€þ x 10 2 4 6 8 u u; ; ; ; : 19A 304 1015A 1015A 470A 75A c2n1 ¼ cosð2n 1Þ d n ¼ 1 2 3 þ þ þ þ ; p 1 x2 = 0 þ 0 32 2 1 2 ð1 þ A Þ ð38Þ " The first two terms have been obtained from Eq. (38) as 24 2 4 6 8 r5 ¼ 9 64 128A 80A 17A A 2 2 2 A c1 ¼ e0 x A þðe2 þ x e1Þu3 þ Oðu Þ; # 3 x2 2 ; 10 2 4 6 8 c3 ¼ p0 þðp2 þ p1Þu3 þ Oðu Þ A 64 þ 160A þ 136A þ 45A þ 5A ; 3 þ þ 1=2 8 ð1 þ A2Þ where 4k Âà 2 2 ; = : e0 ¼ A EðA ÞKðA Þ r6 ¼25A 2r5 3 p"#A 4 2 A2 Solving Eq. (27) for x, we obtain the third approximate fre- 2 3= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ e1 ¼ 3 2 þ A A 2 quency as A 1 þ A2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = 16k b0 þ b1u3 b3u3 þ b5u3u5 2 þ b7u5 2 2 2 2 ; xðAÞ¼x ðAÞ¼ ; ð30Þ e2 ¼e1 þ ½ð2 þ A ÞEðA Þ2ð1 þ A ÞKðA Þ 3 2 = pA3 A b2u3 b4u3 b6u3u5 2 b8u5 3 4 2 2 2 2 ; p0 ¼ 3 ½ð8 þ A ÞKðA Þð8 þ A ÞEðA Þ where u3 and u5 are given in the following 3pA 72 M.A. Razzak "# 8 4 þ 7A2 þ 3A4 Solving Eq. (39), we obtain the second approximate fre- p ¼ 4 þ 5A2 þ A3 5A4=4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 5 2 quency as A 1 þ A rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x e0 þ e2u3; 16k ðAÞ¼ 2ðAÞ¼ ð45Þ p ¼p 8A ½ð32 þ 42A2 þ 7A4ÞEðA2Þ A e1u3 2 1 15pA5 where u is given by Eq. (44). 2ð16 þ 29A2 þ 13A4ÞKðA2Þ: 3 Therefore, the second-order approximate periodic solution Substituting Eq. (37) for Eq. (36) and then equating the of Eq. (33) is coefficients of the terms cos u and cos 3u equal to zeros, x1ðtÞ¼Aðð1 u3Þ cos xt þ u3 cos 3xtÞ; ð46Þ respectively, we obtain where u3 and x respectively, are given by Eqs. (44) and (45). x2 x2 2 ; e0 A þðe2 þ e1Þu3 þ Oðu3Þ¼0 ð39Þ x2 2 : 4. Results and discussion p0 þðp2 þ p1Þu3 þ Oðu3Þ¼0 ð40Þ

At first take u3 = 0, Eq. (39) becomes Based on the modified harmonic balance method (Hosen et al., 2 e0 x A ¼ 0: ð41Þ 2012), an easy approach has been proposed to obtain higher- order approximate frequencies and the corresponding periodic Solving Eq. (41), we obtain solutions for both small and large values of amplitude of the rffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 4k strongly nonlinear oscillators having a rational and an irra- x x 0 2 2 ¼ 1ðAÞ¼ ¼ 1 ½EðA ÞKðA Þ ð42Þ tional force. It has been already mentioned that the determina- A pA2 tion of second-order approximation is very difficult by The first approximate frequency is given in Eq. (42) which methods Nayfeh and Mook (1979), Hu and Tang (2006), was also obtained by Sun et al. (2007). Lim and Wu (2003). In the present article, the higher-order Eliminating x from Eqs. (39) and (40), we obtain approximate frequency as well as periodic solution especially 2 second and third-order has been determined without any com- Ap þðe p Ap e p Þu þðp e p e Þu þ¼0: 0 1 0 2 0 1 3 2 1 1 2 3 plicity and easily analytically solved (Eqs. (24), (30) and (45)). ð43Þ We have calculated the second-order approximate frequen- Solving Eq. (43), we obtain cies for several amplitudes of oscillation (by Eq. (24)) of Eq. (8) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and presented in Table 1. The results obtained in this paper are 2 gðAÞþ g ðAÞþ4Ap0wðAÞ compared with those of others (calculated by Mickens, 2001; u3 ¼ ; ð44Þ 2wðAÞ Lim and Wu, 2003; Tiwari et al., 2005; Ozis and Yildirim, 2007; Ghadimi and Kaliji, 2013; Khan et al., 2012a) all the where g(A)=p1e0 + Ap2 p0e1, w(A)=e1p2 e2p1.

Table 1 Comparison of the approximate frequencies obtained by present method (Eq. (24)) with the exact frequency xe and other existing frequencies (those are obtained by Mickens, 2001; Lim and Wu, 2003; Tiwari et al. (2005), Ozis and Yildirim, 2007; Ghadimi and Kaliji, 2013; Khan et al., 2012a).

A xe Mickens Lim and Wu Tiw. et al. Ozis and Yi Ghadimi and Kaliji Khan et al. Present study

Er(%) Er(%) Er(%) Er(%) Er(%) Er(%) x2 Er(%) x3 Er(%) 0.1 0.084389 0.086280 0.084256 0.086244 0.086268 0.084449 0.0757688 ‘0.084389 0.084389 2.240 0.158 2.198 2.227 0.071 10.215 0.000 0.000 0.2 0.166830 0.170664 0.166563 0.170393 0.170575 0.166964 0.150020 0.166826 0.166830 2.298 0.160 2.136 2.245 0.080 10.076 0.002 0.000 0.4 0.319403 0.327327 0.318863 0.325513 0.326746 0.319757 0.288839 0.319359 0.319407 2.481 0.169 1.913 2.299 0.111 9.569 0.014 0.001 0.6 0.449101 0.461084 0.448326 0.456392 0.459648 0.449777 0.409333 0.448990 0.449109 2.668 0.173 1.624 2.345 0.151 8.855 0.025 0.002 0.8 0.554068 0.569495 0.553140 0.561440 0.567163 0.555136 0.509444 0.553895 0.554076 2.784 0.168 1.331 2.364 0.193 8.054 0.031 0.002 1 0.636780 0.654654 0.635796 0.643594 0.651641 0.638285 0.590597 0.636572 0.636785 2.807 0.155 1.070 2.333 0.236 7.252 0.033 0.000 2 0.847626 0.866026 0.847021 0.850651 0.862895 0.850963 0.811834 0.847505 0.847628 2.171 0.071 0.357 1.801 0.394 4.222 0.042 0.000 3 0.919599 0.933257 0.919328 0.920897 0.931207 0.923295 0.895338 0.919574 0.919614 1.485 0.0295 0.141 1.262 0.402 2.638 0.003 0.002 4 0.950856 0.960769 0.950730 0.951481 0.959428 0.954174 0.933926 0.950862 0.950875 1.043 0.013 0.066 0.902 0.349 1.781 0.000 0.002 5 0.966976 0.974355 0.966913 0.967310 0.973431 0.969779 0.954642 0.966989 0.966992 0.763 0.007 0.035 0.668 0.290 1.276 0.001 0.002 10 0.990916 0.993399 0.990912 0.990954 0.993144 0.992118 0.986940 0.990921 0.990921 0.251 0.000 0.004 0.225 0.121 0.401 0.000 0.000 where Er(%) denotes the absolute percentage error. Harmonic balance method for solving strongly nonlinear oscillators 73

Table 2a Comparison of the approximate frequencies obtained by present method (Eq. (45)) with the exact frequency xe and other existing frequencies (those frequencies obtained by Mickens, 1996a; Ganji et al., 2009; Zhao, 2009; Akbarzade and Farshidianfar, 2014; Khan et al., 2012a) when k ¼ 0:5.

A xe Mickens Ganji et al. Zhao Akbarzade and Farshidianfar Khan et al. Present study Er(%) Er(%) Er(%) Er(%) Er(%) Er(%) 0.1 0.708423 0.707987 0.708424 0.86717 0 0.708431 0.708096 0.708423 2.159 0.000 22.408 0.001 0.046 0.000 0.2 0.712259 0.710582 0.712271 0.870489 0.712390 0.710997 0.712259 0.236 0.001 22.215 0.018 0.177 0.000 0.4 0.726126 0.720330 0.726271 0.882252 0.728011 0.721720 0.726125 0.798 0.020 21.501 0.260 0.607 0.000 0.6 0.745140 0.734651 0.745683 0.897720 0.753326 0.737022 0.745145 1.408 0.073 20.477 1.099 1.090 0.000 0.8 0.765907 0.751536 0.767072 0.913595 0.769064 0.754488 0.765903 1.876 0.152 19.283 0.412 1.491 0.000 1 0.786171 0.769254 0.788075 0.927961 0.790009 0.772287 0.786165 2.152 0.242 18.036 0.488 1.766 0.000 2 0.860447 0.843401 0.864865 0.969782 0.864890 0.843545 0.860451 1.981 0.513 12.707 0.516 1.964 0.000 3 0.899904 0.887017 0.904671 0.984638 0.903592 0.885017 0.899922 1.432 0.530 9.416 0.410 1.654 0.002 4 0.922727 0.912871 0.927153 0.990901 0.925721 0.910077 0.922749 1.068 0.480 7.389 0.325 1.371 0.002 5 0.937317 0.929471 0.941285 0.994030 0.939793 0.926489 0.937338 0.837 0.423 6.051 0.264 1.155 0.002 10 0.968102 0.964358 0.970480 0.998456 0.969373 0.962054 0.968113 0.387 0.246 3.136 0.131 0.625 0.001

Table 2b Comparison of the approximate frequencies obtained by present method (Eq. (45)) with the exact frequency xe and other existing frequencies when k ¼ 0:75.

A xe Mickens Ganji et al. Zhao Akbarzade and Farshidianfar Khan et al. Present study Er(%) Er(%) Er(%) Er(%) Er(%) Er(%) 0.1 0.502786 0.501865 0.502788 0.664804 0.502805 0.502095 0.502786 0.183 0.000 32.224 0.003 0.138 0.000 0.2 0.510841 0.507336 0.510876 0.674494 0.511126 0.508213 0.510840 0.686 0.007 32.036 0.056 0.515 0.000 0.4 0.539214 0.527553 0.539633 0.708046 0.543139 0.530449 0.539211 2.163 0.0778 31.311 0.728 1.626 0.000 0.6 0.576587 0.556389 0.577983 0.750517 0.592663 0.561244 0.576575 3.503 0.242 30.165 2.788 2.661 0.002 0.8 0.615781 0.589244 0.618545 0.792449 0.655744 0.595194 0.615756 4.310 0.449 28.6901 6.490 3.343 0.004 1 0.652771 0.622597 0.656958 0.829156 0.660432 0.628649 0.652735 4.622 0.642 27.021 1.174 3.695 0.005 2 0.780662 0.752986 0.788662 0.930630 0.788702 0.753788 0.780668 3.545 1.025 19.211 1.030 3.443 0.000 3 0.844964 0.824742 0.853020 0.965092 0.851304 0.821921 0.845013 2.393 0.954 14.217 0.750 2.727 0.006 4 0.881255 0.866025 0.888492 0.979408 0.886250 0.861890 0.881313 1.728 0.821 11.138 0.567 2.197 0.007 5 0.904141 0.892119 0.910509 0.986516 0.908195 0.887659 0.904197 1.330 0.704 9.111 0.448 1.823 0.006 10 0.951696 0.946033 0.955378 0.996522 0.953690 0.942570 0.951722 0.595 0.387 4.710 0.210 0.959 0.003

results together with exact frequency have been shown in Mickens, 2001; Lim and Wu, 2003; Tiwari et al., 2005; Ozis Table 1. Table 1 indicates that the approximate frequencies and Yildirim, 2007; Ghadimi and Kaliji, 2013; Khan et al., (concern of this paper) are better than those obtained by 2012a. 74 M.A. Razzak

Table 2c Comparison of the approximate frequencies obtained by present method (Eq. (45)) with the exact frequency xe and other existing frequencies when k ¼ 0:95.

A xe Mickens Ganji et al. Zhao Akbarzade and Farshidianfar Khan et al. Present study Er(%) Er(%) Er(%) Er(%) Er(%) Er(%) 0.1 0.231367 0.228836 0.231391 0.323516 0.231436 0.229483 0.231367 1.094 0.011 39.828 0.023 0.814 0.000 0.2 0.252549 0.243639 0.252836 0.354238 0.253476 0.246025 0.252547 3.528 0.114 40.265 0.367 2.583 0.000 0.4 0.317642 0.293022 0.319674 0.447114 0.327109 0.300135 0.317594 7.751 0.640 40.760 2.981 5.511 0.015 0.6 0.391035 0.354195 0.395577 0.547084 0.422197 0.364869 0.390906 9.421 1.162 39.907 7.967 6.692 0.033 0.8 0.459947 0.416090 0.466860 0.634908 0.527257 0.428086 0.459761 9.535 1.503 38.039 14.634 6.927 0.0404 1 0.520335 0.473633 0.529168 0.706124 0.534618 0.485141 0.520138 8.975 1.697 35.706 2.745 6.764 0.038 2 0.709629 0.671950 0.721931 0.886069 0.721987 0.674087 0.709623 5.310 1.734 24.864 1.742 5.009 0.000 3 0.797913 0.771310 0.809330 0.943366 0.807038 0.768077 0.798006 3.334 1.431 18.229 1.144 3.739 0.012 4 0.846399 0.826640 0.856307 0.966748 0.853359 0.821520 0.846507 2.335 1.171 14.219 0.822 2.940 0.0123 5 0.876561 0.861071 0.885117 0.978276 0.882101 0.855466 0.876659 1.767 0.976 11.604 0.632 2.407 0.011 10 0.938333 0.931114 0.943122 0.994413 0.940956 0.926722 0.938376 0.7693 0.510 5.977 0.280 1.238 0.005

6 Next, we have calculated the second-order approximate fre- 4 quency (by Eq. (45)) of Eq. (33) with several amplitudes of oscillation and some different values of k that are compared 2 with numerical result and other existing results (those results x 0 01234567 obtained by Mickens, 1996; Ganji et al., 2009; Zhao, 2009; -2 Akbarzade and Farshidianfar, 2014; Khan et al., 2012a) which -4 are presented in Tables 2a–c. The absolute percentage errors of -6 t each method has been calculated and presented in all Tables 1 and 2c. The results of these Tables show that the approximate Figure 1b The present method solution of Eq. (8) has been frequencies (concern by this paper) provide better results than presented (denoting b circles) when large value of amplitude those obtained by Mickens (1996), Ganji et al. (2009), Zhao A = 5.0, with initial conditions [xð0Þ¼5:0; x_ð0Þ¼0]. Corre- (2009), Akbarzade and Farshidianfar (2014), Khan et al. sponding numerical solution has been presented (denoted by solid (2012a). Thus, the present method provides better result than line) to compare with present method. other existing results.

0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2

x 0

x 0 -0.2 0123 4 5 6 7 8 9 10 11 12 -113579111315 -0.2 -0.4 -0.4 -0.6 -0.8 -0.6 t t -0.8 Figure 2a The present method solution of Eq. (33) has been Figure 1a The present method solution of Eq. (8) has been presented (denoting by circles) when small value of amplitude presented (denoting by circles) when small value of amplitude A = 0.6 and k ¼ 0:75, with initial conditions A = 0.6, with initial conditions [xð0Þ¼0:6; x_ð0Þ¼0]. Corre- [xð0Þ¼0:6; x_ð0Þ¼0]. Corresponding numerical solution has sponding numerical solution has been presented (denoted by solid been presented (denoted by solid line) to compare with present line) to compare with present method. method. Harmonic balance method for solving strongly nonlinear oscillators 75

6 Seeing all the figures, we observe that the present method 4 solutions are nicely agreement with the corresponding numer-

2 ical solutions for both small and large amplitudes of oscillations.

x 0 012345678 -2 5. Conclusions -4 -6 t In present work, a simple harmonic balance approach has been presented to obtain higher-order approximations of strongly Figure 2b The present method solution of Eq. (33) has been nonlinear oscillator systems having a rational and an irrational presented (denoting by circles) when small value of amplitude force. Recently, many analytical techniques have been devel- k : A = 5.0 and ¼ 0 75, with initial conditions oped to solve strongly nonlinear oscillators. But these tech- : ; _ [xð0Þ¼5 0 xð0Þ¼0]. Corresponding numerical solution has been niques are very difficult to determine higher-order presented (denoted by solid line) to compare with present method. approximations because a set of complex nonlinear algebraic equations involve higher order terms and it is very difficult to solve these equations analytically. In this article, this limita- tion has been eliminated by applying a simple factor to the 0.8 nonlinear oscillators. The solution procedure of the present 0.6 approach is very simple involving a lower order term. Results 0.4 0.2 obtained in this paper are compared with other existing results. As indicated, the error of the present method is much low than x 0 -0.2-11 3 5 7 9 11131517 others. On the other hand, it makes the approximate solution -0.4 rapidly converge. Thus, the present approach is an extremely -0.6 effective and powerful method for solving strongly nonlinear -0.8 oscillator systems arising in nonlinear science and engineering t especially in vibration engineering.

Figure 2c The present method solution of Eq. (33) has been Acknowledgments presented (denoting by circles) when small value of amplitude A = 0.6 and k ¼ 0:95, with initial conditions The author is grateful to the reviewers for their helpful com- [xð0Þ¼0:6; x_ð0Þ¼0]. Corresponding numerical solution has been ments/suggestions in improving the manuscript. presented (denoted by solid line) to compare with present method.

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University of Bahrain Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com

ORIGINAL ARTICLE Optimal homotopy asymptotic method for solving nth order linear fuzzy initial value problems

A.F. Jameel a,*, A.I.M. Ismail a, F. Mabood b a School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia b Department of Mathematics, University of Peshawar, Pakistan

Received 25 September 2014; revised 28 March 2015; accepted 25 April 2015 Available online 3 July 2015

KEYWORDS Abstract In this paper the optimal homotopy asymptotic method (OHAM) is employed to obtain Fuzzy numbers; approximate analytical solution of nth order ðn P 2Þ linear fuzzy initial value problems (FIVPs). Fuzzy differential equations; The convergence theorem of this method in fuzzy case is presented and proved. This method pro- Optimal homotopy vides us with a convenient way to control the convergence of approximation series. The method is asymptotic method tested on nth linear FIVPs and comparisons of the exact solution that were made with numerical results showed the effectiveness and accuracy of this method. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction 2015). Approximate-analytical methods such as the Adomian Decomposition Method (ADM), Homotopy Perturbation Many dynamical real life problems may be formulated as a Method (HPM) and Variational Iteration Method (VIM) have mathematical model. These problems can be formulated either been used to solve fuzzy initial value problems involving ordi- as a system of ordinary or partial differential equations. Fuzzy nary differential equations. Ghanbari (2009) utilized HPM to differential equations are a useful tool to model a dynamical solve first order linear fuzzy initial value problems. The ADM system when information about its behavior is inadequate. was employed by Babolian et al. (2004) and Allahviranlo et al. Fuzzy ordinary differential equations may arise in the mathe- (2008) to solve first order linear and nonlinear fuzzy initial value matical modeling of real world problems in which there is problems. Abbasbandy et al. (2011) used the VIM to solve linear some uncertainty or vagueness. Fuzzy initial value problems systems of first order fuzzy initial value problem. (FIVPs) appear when the modeling of these problems was OHAM is somewhat different from other approximate- imperfect and its nature is under uncertainty. Fuzzy initial analytical methods in that it gives extremely good results for value problems arise in several areas of mathematics and even a large domain with minimal terms of the approximate science including population models (Ahmad and De Baets, series solution. In OHAM, the control and adjust of the con- 2009; Omer and Omer, 2013), mathematical physics (El vergence region are provided in a convenient way. Moreover, Naschie, 2005) and medicine (Abbod et al., 2001; Barro and OHAM is also parameter free and provides better accuracy Marin, 2002) and other applications (Sahu and Saha Ray, over the approximate analytical methods at the same order of approximation. OHAM was introduced recently by Marinca et al. (2008) * Corresponding author. E-mail address: [email protected] (A.F. Jameel). and applied for solving nonlinear problems without depending on the small parameter (Herisanu et al., 2008; Marinca and Peer review under responsibility of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2015.04.004 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 78 A.F. Jameel et al.

Herisanu, 2008; Marinca et al., 2009). Mabood (2014) has pro- In this paper the class of all fuzzy subsets of R will be vided a comparative study between OHAM and HPM for denoted by E~ and satisfy the following properties (Dubois and strongly nonlinear equation. In this study, it was observed that Prade, 1982; Mansouri and Ahmady, 2012): OHAM gives an accurate solution as compared to HPM.

Moreover, an advantage of OHAM is that it does not need 1. lðtÞ is normal, i.e. 9 t0 2 R with lðt0Þ¼1. any initial guess or to identify the h-curve like Homotopy 2. lðtÞis convex fuzzy set, i.e. lðkt þð1 kÞsÞ P minflðtÞ; Analysis Method (HAM) and it is also parameter free. lðsÞg 8t; s 2 R; k 2½0; 1. Furthermore, the OHAM has built in convergence criteria 3. l upper semi-continuous on R, and ft 2 R : lðtÞ > 0g is similar to HAM but with a greater degree of flexibility compact. (Iqbal et al., 2010). The proposed method (OHAM) has also been successfully applied on various engineering problems E~ is called the space of fuzzy numbers and R is a proper (Alomari et al., 2013a,b; Anakira et al., 2013; Mabood et al., subset of E~. 2013a,b; 2014a,b; Herisanu et al., 2015; Marinca and

Herisanu, 2014). Define the r-level set x 2 R, ½lr ¼fx n lðxÞ P rg; 6 6 In this paper, our aim is to apply OHAM to nth order 0 r 1 where ½l0 ¼fx n lðxÞ > 0g is compact which is a n P 2 FIVP directly without reducing it to a system. To ð Þ closed bounded interval and denoted by ½lr ¼ðlðtÞ; lðtÞÞ.In the best of our knowledge, this is the first attempt at solving the parametric form, a fuzzy number is represented by an ordered a high order FIVP using the OHAM with proof of the conver- pair of functions ðlðtÞ; lðtÞÞ, which satisfies (Kaleva, 1987): gence in fuzzy case. The outline of this paper is as follows. We will start in Section 2 with some preliminary concepts about 1. lðtÞ is a bounded left continuous non-decreasing function fuzzy numbers. In Section 3 we reviewed the concept of over ½0; 1. OHAM and formulated it to obtain a reliable approximate 2. lðtÞ is a bounded left continuous non-increasing function solution to nth order FIVPs. In Section 4, the convergence the- over ½0; 1. orem of OHAM is presented and proved, In Section 5, we con- 3. lðtÞ 6 lðtÞ; r 2½0; 1. sider numerical examples to show the capability of this method and finally, in Section 6, we give the conclusions of this study. A crisp number r is simply represented by lðrÞ ¼ lðrÞ¼r, r 2. Preliminaries 2½0; 1. ~ Definition 2.1 (Bodjanova, 2006). The r-level (or r-cut) set of a Definition 2.3 (Seikkala, 1987).IfE be the set of all fuzzy numbers, we say that fðtÞ is a fuzzy function if f : R ! E~. fuzzy set A~, labeled as A~r, is the crisp set of all x 2 X such that lA~ P r i.e. Definition 2.4 (Fard, 2009). A mapping f : T ! E~ for some interval T # E~ is called a fuzzy function process and we denote ~ Ar ¼ fgx 2 XjlA~ > r; r 2½0; 1 r-level set by:

~ Definition 2.2. Fuzzy numbers are a subset of the real numbers ½fðtÞr ¼½fðt; rÞ; fðt; rÞ; t 2 T; r 2½0; 1 set, and represent uncertain values. Fuzzy numbers are linked to degrees of membership which state how true it is to say if some- The r-level sets of a fuzzy number are much more effective thing belongs or not to a determined set. A fuzzy number (Dubois as representation forms of fuzzy set than the above. Fuzzy sets and Prade, 1982) l is called a triangular fuzzy number if defined can be defined by the families of their r-level sets based on the by three numbers a < b < c where the graph of lðxÞ is a triangle resolution identity theorem. with the base on the interval [a,b]andvertexatx ¼ b,andits Definition 2.5R (Kaleva, 1987). The fuzzy integral of fuzzy pro- membership function has the following form: (see Fig. 1) ~ b ~ cess, fðt; rÞ; a fðt; rÞdt for a; b 2 T and r 2½0; 1 is defined by: 8 Z Z Z > 0; if x < a b b b <> ~ xa ; if a 6 x 6 b fðt; rÞdt ¼ fðt; rÞdt; fðt; rÞdt lðx; a; b; cÞ¼ ba a a a > cx ; if b 6 x 6 y :> cb 0; if x > c Definition 2.6 (Zadeh, 1965). Each function f : X ! Y induces another function f~: FðXÞ!FðYÞ defined for each fuzzy and its r-level is ½l ¼½a þ rðb aÞ; c rðc bÞ; r 2½0; 1. r interval U in X by: Sup 1 UðxÞ; if y 2 rangeðfÞ μ(x) f~ðUÞðyÞ¼ x2f ðyÞ 0; if y R rangeðfÞ 1 This is called the Zadeh extension principle. 0.5 Definition 2.7 (Salahshour, 2011). Consider x~; y~ 2 E~. If there 0 ~ α β γ x exists z~ 2 E such that x~ ¼ y~ þ z~, then z~ is called the H-difference (Hukuhara difference) of x and y and is denoted Figure 1 Triangular fuzzy number. by z~ ¼ x~€y~. Optimal homotopy asymptotic method 79

~ ~ Definition 2.8 (Zadeh, 2005).Iff : I ! E and y0 2 I, where 3. Fuzzification and defuzzification of OHAM ~ I 2½t0; T. We say that f Hukuhara Differentiable at y0, if there ~0 ~ exists an element ½f r 2 E such that for all h > 0 sufficiently The general structure of OHAM for solving crisp nth order ~ €~ ~ €~ ordinary differential equations was described in Gupta and small (near to 0), exists fðy0 þ h;rÞ fðy0;rÞ, fðy0;rÞ fðy0 h;rÞ and the limits are taken in the metric ðE~;DÞ Saha Ray (2014), Herisanu et al. (2015), Marinca and Herisanu (2014). To solve the nth order FIVP, there is a need to fuzzify and then defuzzify OHAM. Consider the following f~ðy þ h; rÞ€f~ðy ; rÞ f~ðy ; rÞ€f~ðy h; rÞ lim 0 0 ¼ lim 0 0 general nth order FIVP h!0þ h h!0þ h y~ðnÞ t ft~ ; y~ t ; y~ 0 t ; y~ 00 t ; ...y~ðn1Þ t w~ t ; t t ; T ~0 ð Þ¼ ð Þ ð Þ ð Þ ð Þ þ ð Þ 2½0 The fuzzy set ½ f ðy0Þr is called the Hukuhara derivative of ~0 ð1Þ ½f r at y0. subject to the initial conditions These limits are taken in the space ðE~; DÞ if t0 or T, then we 0 0 ðn1Þ ðn1Þ consider the corresponding one-side derivation. Recall that y~ðt0Þ¼y~0; y ðt0Þ¼y~0; ...y ðt0Þ¼y~0 ð2Þ x~€y~ ¼ z~ 2 E~ are defined on r-level set, where ½x~ €½y~ ¼½z~ , r r r where y~ is a fuzzy function of the crisp variable t with f~being a 8r 2½0; 1. By consideration of definition of the metric D all fuzzy function of the crisp variable t, the fuzzy variable y~ the r-level sets ½ f~ð0Þ are Hukuhara differentiable at y , with r 0 and the fuzzy Hukuhara-derivatives y~ 0ðtÞ; y~ 00ðtÞ; ...y~ðn1ÞðtÞ. ~0 ~ ~ Hukuhara derivatives ½ f ðy0Þr, when f : I ! E is Hukuhara Here y ðnÞ is the fuzzy nth order Hukuhara-derivative and differentiable at y with Hukuhara derivative ½ f~0ðy Þ and it 0 ðn1Þ 0 0 r y~ðt0Þ; y~ ðt0Þ; ...y~ ðt0Þ are convex fuzzy numbers. We denote ~ leads to that f is Hukuhara differentiable for all r 2½0; 1 which the fuzzy function y by y~ ¼½y; y for t 2½t0; T and r 2½0; 1.It satisfies the above limits i.e. if f is differentiable at means that the r-level set of y~ðtÞ can be defined as: ~0 hi t0 2½t0 þ a; T then all its r-levels ½ f ðtÞr are Hukuhara differ- ½y~ðtÞ ¼ yðt; rÞ; yðt; rÞ entiable at t0. r hi Definition 2.9 (Salahshour, 2011). Define the mapping 0 0 0 ðn1Þ ½y~ ðtÞr ¼ y ðt; rÞ; y ðt0; rÞ ; ...½y~ ðtÞr ~0 ~ ~0 hi f : I ! E and y0 2 I, where I 2½t0; T. We say that f is ðn1Þ ðn1Þ Hukuhara differentiable t 2 E~, if there exists an element ¼ y ðt; rÞ; y ðt; rÞ ~ðnÞ ~ ½ f r 2 E such that for all h > 0 sufficiently small (near to 0), hi ~ðn1Þ €~ðn1Þ ~ðn1Þ €~ðn1Þ exist f ðy0 þ h; rÞ f ðy0; rÞ,andf ðy0; rÞ f ½y~ðt0Þr ¼ yðt0; rÞ; yðt0; rÞ ; hi y h r E~ D ð 0 ; Þ and the limits are taken in the metric ð ; Þ 0 0 0 ðn1Þ ½y~ ðt0Þ ¼ y ðt0; rÞ; y ðt0; rÞ ; ...; ½y~ ðt0Þ r hir n 1 n 1 f~ð Þðy þ h; rÞ€f~ð Þðy ; rÞ ðn1Þ ðn1Þ lim 0 0 ¼ y ðt0; rÞ; y ðt0; rÞ h!0þ h ~ðn1Þ €~ðn1Þ where the fuzzy inhomogeneous term is ½w~ðtÞ ¼ f ðy0; rÞ f ðy0 h; rÞ r ¼ lim w t; r ; w t; r . h!0þ h ½ ð Þ ð Þ Since y~ðnÞðtÞ¼ft~ ; y~ðtÞ; y~ 0ðtÞ; y~ 00ðtÞ; ...y~ðn1ÞðtÞ þ w~ðtÞ. ~ðnÞ exists and equal to f and for n ¼ 2 we have second order Let Hukuhara derivative. YðtÞ¼y~ðtÞ; y~ 0ðtÞ; y~ 00ðtÞ; ...y~ðn1ÞðtÞ; such that ~ Theorem 2.1 Mansouri and Ahmady, 2012. Let f : ½t0 þ a;T! ~ t;r t;r ; t;r E~ be Hukuhara differentiable denoted by Yð Þ¼½Yð Þ Yð Þ 0 ðn1Þ 0 ðn1Þ hihi ¼½yðt;rÞ;y ðt;rÞ;...;y ðt;rÞ;yðt;rÞ;y ðt;rÞ;...;y ðt;rÞ ~0 0 0 0 0 ½ f ðtÞr ¼ f ðtÞ; f ðtÞ ¼ f ðt; rÞ; f ðt; rÞ r Also we can write Then the boundary functions f 0ðt; rÞ; f0ðt; rÞ are differentiable ~ ½fðt; y~Þr ¼½fðt; y~; rÞ; fðt; y~; rÞ ð3Þ hi ~0 0 0 By using Zadeh extension principles as mentioned in Zadeh ½ f ðtÞr ¼ðfðt; rÞÞ ; ðfðt; rÞÞ ; 8r 2½0; 1 (2005), we have ~ Theorem 2.2 Xiaobin and Dequan, 2013. Let f : ½t0 þ a; T! f~ðt; Yð~ t; rÞÞ ¼ ½fðt; Yð~ t; rÞÞ; fðt; Yð~ t; rÞÞ; ; such that E~ be Hukuhara differentiable denote by hihi f t; ~ t; r t; t; r ; t; r t; ~ t; r ~0 0 0 0 0 ð Yð ÞÞ ¼ Fð Yð Þ Yð ÞÞ¼Fð Yð ÞÞ ½ f ðtÞr ¼ f ðtÞ; f ðtÞ ¼ f ðt; rÞ; f ðt; rÞ r ~ ~ When the boundary functions f 0ðt; rÞ; f0ðt; rÞ are differen- fðt; Yðt; rÞÞ ¼ Gðt; Yðt; rÞ; Yðt; rÞÞ ¼ Gðt; Yðt; rÞÞ tiable we can write for nth order fuzzy derivative Then we have hi ~ðnÞ ðnÞ 0 ðnÞ 0 y ðnÞðt; rÞ¼F t; Yð~ t; rÞ þ wðt; rÞð4Þ ½f ðtÞr ¼ðf ðt; rÞÞ ; ðf ðt; rÞÞ ; 8r 2½0; 1 80 A.F. Jameel et al. ðnÞ ~ Xn y ðt; rÞ¼G t; Yðt; rÞ þ wðt; rÞð5Þ ~ i ½;ðt; p; CiðrÞr ¼ y~0ðt; rÞþ ½y~iðt; CiðrÞÞrp ð16Þ where the membership function of Fðt; Yð~ t; rÞÞ þ wðt; rÞ and i¼1 Gðt; Yð~ t; rÞÞ þ wðt; rÞ can be defined as Now substitute Eq. (14) into (4) and (5), and equating the coefficients of like powers of p, the following linear equations t ~ t r w t r min y~ðnÞ t ~ r ~ t r Fð ; Yð ; ÞÞ þ ð ; Þ¼ ð ; lð ÞÞ : ljl 2½Yð ; Þr are obtained. The zeroth order problem is given by (12), and the first and second order problems are given as follows. ~ ðnÞ ~ Gðt; Yðt; rÞÞ þ wðt; rÞ¼max y~ ðt; l~ðrÞÞ : ljl 2½Yðt; rÞr First order problem: for all r 2½0; 1, Eqs. (4) and (5) can be written as follows ( Lnðy1ðt; rÞÞ wðt; rÞ¼C1ðrÞF 0ðy~0ðt; rÞÞ ~ ð17Þ Lnðyðt; rÞÞ wðt; rÞFðt; Yðt; rÞÞ ¼ 0 ð6Þ Lnðy1ðt; rÞÞ wðt; rÞ¼C1ðrÞG0ðy~0ðt; rÞÞ @½y B yðt; rÞ; r ¼ 0 ð7Þ @½y~ @t B y~ ðt; rÞ; 1 r ¼ 0 ð18Þ 1 @t and for the upper bound we have Second order problem 8 Lnðyðt; rÞÞ wðt; rÞGðt; Yð~ t; rÞÞ ¼ 0 ð8Þ >Lnðy2ðt;rÞÞ Lnðy1ðt;rÞÞ ¼ C2ðrÞF 0ðy~0ðt;rÞÞ <> @½y þC1ðrÞ½Lnðy1ðt;rÞÞþF1ðy~0ðt;rÞ;y~1ðt;rÞÞ B yðt; rÞ; r ¼ 0 ð9Þ > ~ @t :>Lnðy2ðt;rÞÞ Lnðy1ðt;rÞÞ ¼ C2ðrÞG0ðy0ðt;rÞÞ According to OHAM described in Marinca et al. (2008), Eq. þC1ðrÞ½Lnðy1ðt;rÞÞþG1ðy~0ðt;rÞ;y~1ðt;rÞÞ (1) can be written as follows: ð19Þ 1 p t; p w t; r ð Þ Lnð½;ð ÞrÞ ð Þ @½y~ hi B y~ ðt; rÞ; 2 r ¼ 0 ð20Þ ~ 2 @t ¼ Hðp; rÞ Ln ½;ðt; pÞr wðt; rÞF ½;ðt; pÞr ð10Þ The general nth order formula with respect to y~nðt; rÞ is given ð1 pÞ L ð½;ðt; pÞ Þwðt; rÞ by: n hir ~ 8 ¼ Hðp; rÞ Lnð½;ðt; pÞrÞwðt; rÞ Gð½;ðt; pÞrÞ ð11Þ >Lnðynðt;rÞÞ Lnðyn1ðt;rÞÞ ¼ CnðrÞF 0ðy~0ðt;rÞÞ > "# ! ! > Xn1 Xn1 ~ > @½;ðt; pÞ > þ CiðrÞ Lnðyn iðt;rÞÞþ F n i yjðt;rÞ B½;ð~ t; pÞ ; r ¼ 0 ð12Þ < r @t i¼1 j¼0 > ~ >Lnðynðt;rÞÞ Lnðyniðt;rÞÞ ¼ Cn"#ðrÞG0ðy0ðt;rÞÞ ! @n½;ðt;pÞ n > where r, @ ½;ðt;pÞr are the linear operators of > Xn1 Xn1 Ln ¼ @tn Ln ¼ @tn > :> þ CiðrÞ Lnðyniðt;rÞÞþ Gni yjðt;rÞ Eqs. (10) and (11) respectively, p 2½0; 1 is an embedding i¼1 j¼0 ~ parameter, and Hðp; rÞ is a nonzero auxiliary fuzzy function, ð21Þ ~ ~ for p „ 0 and Hðp; rÞ =0,½;ðt; pÞr is an unknown fuzzy func- tion, respectively. When p = 0 and p = 1, we get: @½y~n B y~ ðt; rÞ; r ¼ 0 ð22Þ n @t ½;ðt; 0Þr ¼ y0ðt; rÞ; ½;ðt; 1Þr ¼ yðt; rÞ ð13Þ P P n1 n1 ½;ðt; 0Þr ¼ y0ðt; rÞ; ½;ðt; 1Þr ¼ yðt; rÞ where F j¼0 y~jðt; rÞ and G j¼0 y~jðt; rÞ are the coefficient ~ n ~ ~ Thus, as p increases from 0 to 1, the solution ½;ðt; pÞr varies p of in the expansion of F½;ðt; pÞr and G½;ðt; pÞr about the from y~0ðt; rÞ to y~ðt; rÞ, where y~ðt; rÞ is obtained from Eq. (1) embedding parameter p for p ¼ 0 we have: 8 "# !! ! > Xn X1 Xn > ~ n > F ; t; p; CiðrÞ ¼F0ðy~0ðt; rÞÞ þ Fn ½y~i p @½y~0r < r L~ ðy~ ðt; rÞÞ þ w~ðt; rÞ¼0; B y~ ðt; rÞ; ¼ 0 ð14Þ i¼1 n¼1 i¼0 n 0 0 !!r ! @t > > Xn X1 Xn > ~ n Choose auxiliary function Hð~ p; rÞ for Eqs. (3) and (4) in the : G½; t; p; CiðrÞ ¼G0ðy~0ðt; rÞÞ þ Gn ½y~ir p form: i¼1 r n¼1 i¼0 8 ð23Þ > Xn > 2 i <> Hðp; rÞ¼C1ðrÞp þ C2ðrÞp þ¼ CiðrÞp The convergence of the series (10) depends upon the auxiliary i¼1 ð15Þ fuzzy constants C~ ðrÞ; C~ ðrÞ; ..., then at p ¼ 1, we obtain: > Xn 1 2 > 2 i : Hðp; rÞ¼C1ðrÞp þ C2ðrÞp þ¼ CiðrÞp ! "# ! i¼1 Xn Xn Xn ~ ~ y~ t; CiðrÞ; r ¼ y~0ðt; rÞþ y~i t; CiðrÞ ð24Þ where C1ðrÞ; C2ðrÞ; ... are the constants that become function i¼1 i¼1 i¼1 r of r to be determined depending on the value of r for all. ~ Expanding ½;ðt; p; CiðrÞÞr about p, we obtain the approximate Substituting (24) into (21) results in the following solution series: residual: Optimal homotopy asymptotic method 81 8 ! !! !! > Xn Xn Xn > As mentioned in Section 2 all fuzzy sets are subsets of R and >R t; C ðrÞ;r ¼ L y t; C ðrÞ;r wðt;rÞF y~ t; C~ ðrÞ;r < i n i i the r-level sets are crisp sets, and then we can define the i¼1 ! i¼1 !! i¼1 !! hi > Xn Xn Xn ~ 1 1 1 > ~ sequence Smðt; rÞ ¼ fgSmðt; rÞ ; Smðt; rÞ as :>R t; CiðrÞ;r ¼ Ln y t; CiðrÞ;r wðt;rÞG y~ t; CiðrÞ;r m¼0 m¼0 m¼0 i¼1 i¼1 i¼1 follows. ð25Þ For the lower bound of Eq. (30) we have ~ If R¼0, then y~ yields the exact solution but mostly for nonlinear problems which does not happen in general. To S ðt; rÞ¼y ðt; rÞ ~ 0 0 determine the auxiliary fuzzy constants of CiðrÞ, i =1,2...n, ~ we choose t0 and T such that optimum values of CiðrÞ for S1ðt; rÞ¼y0ðt; rÞþy1ðt; rÞ the convergent solution of the desired problem is obtained. ~ To find the optimal values of CiðrÞ for each r-level set here, S2ðt; rÞ¼y0ðt; rÞþy1ðt; rÞþy2ðt; rÞ we apply the least squares method (Mabood et al., 2013a)as follows: ... !Z ! Xn T Xn ~ 2 Smðt; rÞ¼y0ðt; rÞþy1ðt; rÞþy2ðt; rÞþymðt; rÞ S t; C~iðrÞ; r ¼ R t; C~iðrÞ; r dt ð26Þ t i¼1 0 i¼1 And for the upper bound of Eq. (30) we have where R~ is the residual, ( S0ðt; rÞ¼y0ðt; rÞ ½Rr ¼ Lnð½y Þwðt; rÞFð½y~rÞ r ð27Þ S1ðt; rÞ¼y0ðt; rÞþy1ðt; rÞ ½Rr ¼ Lnð½yrÞwðt; rÞGð½y~rÞ and S2ðt; rÞ¼y0ðt; rÞþy1ðt; rÞþy2ðt; rÞ

@S~ @S~ @S~ ... ¼ ¼ ¼ 0 ð28Þ @C~ ðrÞ @C~ ðrÞ @C~ ðrÞ 1 2 n Smðt; rÞ¼y0ðt; rÞþy1ðt; rÞþy2ðt; rÞþymðt; rÞ where t and T being the end points of Eq. (1) to locate the 0 According to Gupta and Saha Ray (2014) we have to show desired C~ ðrÞ (i =1,2,...,n). The convergence of the nth 1 i that fS~ ðt; rÞg is a Cauchy sequence in the Hilbert space R. approximate solution depends upon unknown constants m m¼0 We start with the lower bound of Eq. (20). Consider C~ ðrÞ. With these constants known, the approximate solution i (of order n) is well-determined. Thus after substituting the 6 ~ kkSmþ1ðt; rÞSmðt; rÞ ¼ ymþ1ðt;rÞ r ymðt; rÞ determined constants CiðrÞ in Eq. (24) the approximate solu- tion of Eq. (1) can be written in the following form 6 2 6 6 mi01 ! r ym1ðt; rÞ r yi0 ðt; rÞ Xn Xn1 ~ y~ t; CiðrÞ; r ¼ y~iðt; rÞð29Þ Now for each r 2½0; 1, n:m 2 N and m P n P i0 i¼1 i¼0 It has been proved that Eq. (1) has a unique fuzzy solution kkSmðt; rÞSnðt; rÞ ¼ðk Smðt; rÞSm1ðt; rÞÞ in each case of r-level set for all r 2½0; 1 (Mansouri and Ahmady, 2012). þ ðkSm1ðt; rÞSm2ðt; rÞÞþþðSnþ1ðt; rÞSnðt; rÞÞ

6 kkSmðt; rÞSm1ðt; rÞ þ kkSm1ðt; rÞSm2ðt; rÞ þ 4. OHAM convergence in fuzzy environments 6 mi0 þ kkSnþ1ðt; rÞSnðt; rÞ þr yi0 ðt; rÞ In this section, we introduce the convergence of the solution mi01 ni0þ1 of the nth order FIVP (1) by OHAM in Section 3. þ r yi0 ðt; rÞ þþr yi0 ðt; rÞ According to Theorem 2 in Gupta and Saha Ray (2014),we 1 rmn define the following theorem. ¼ rni0þ1y ðt; rÞ 1 r i0

Theorem 4.1. Let the solution components y~0ðt; rÞ, y~1ðt; rÞ, This implies limm;n!1kkSmðt; rÞSnðt; rÞ ¼ 0 (since y~ ðt; rÞ,... be defined as given in Eq. (14) and Eqs. (17)-(22). 1 2 P 0 < r < 1). Therefore, fSmðt; rÞgm¼0 is a Cauchy sequence in n1 The series solution i¼0 y~iðt; rÞ as defined in Eq. (29) the Hilbert space R. converges if there exist 0 < r < 1 such that y~ ðt; rÞ 6 iþ1 Similarly for the upper bound of Eq. (30) we consider ry~iðt; rÞ8i P i0 for some i0 2 N. S ðt; rÞS ðt; rÞ ¼ kky ðt; rÞ 6 rkky ðt; rÞ Proof. According to Section 2 the defuzzification of Eq. (27) mþ1 m mþ1 m 6 r2kky ðt; rÞ 6 for all r 2½0; 1 is given by m1 6 mi01 "# r yi0 ðt; rÞ Xn1 Xn1 Xn1 y~iðt; rÞ¼ yiðtÞ; yiðtÞ ¼½yiðt; rÞ; yiðt; rÞ ð30Þ Now for each r 2½0; 1, n:m 2 N and m P n P i0 i¼0 i¼0 i¼0 r 82 A.F. Jameel et al. 00 0 S ðt; rÞS ðt; rÞ y ðtÞ¼4y ðtÞ4yðtÞþ4t 4; t P 0 ð31Þ m n ¼ðS ðt; rÞS ðt; rÞÞ þ ðS ðt; rÞS ðt; rÞÞþ m m1 m1 m2 0 yð0Þ¼ð2 þ r; 4 rÞ; y ð0Þ¼ð3 þ 2r; 9 2rÞ þ ðS ðt; rÞS ðt; rÞÞ nþ1 n 6 S ðt; rÞS ðt; rÞ þ S ðt; rÞS ðt; rÞ þ 8r 2½0; 1 m m1 m1 m2 þ S ðt; rÞS ðt; rÞ The exact solution of Eq. (31) was given in Xiaobin and nþ1 n m i m i 1 6 þr 0 y ðt; rÞ þ r 0 y ðt; rÞ þ Dequan (2013). According to Section 3 we can construct fifth i0 i0 mn order OHAM series as follows. n i 1 1 r n i 1 þ r 0þ y ðt; rÞ ¼ r 0þ y ðt; rÞ i0 1 r i0 Zeroth order problem: This implies lim kS ðt;rÞS ðt;rÞk¼ 0(since0< r < 1). y~00ðt; rÞ¼4t 4 m;n!1 m n i ¼ 0 ð32Þ 1 0 Therefore, fSmðt;rÞgm¼0 is a Cauchy sequence in the Hilbert y~0ð0; rÞ¼½2 þ r; 4 ry~0ð0; rÞ¼½3 þ 2r; 9 2r ~ 1 space R. Thus fSmðt;rÞgm¼0 is a Cauchy sequence in the Hilbert space R and hence the series solution First–fifth order problems hiP n1 h ! ! ! yiðt;rÞ;yiðt;rÞ ¼ i¼0 yiðt;rÞ converges for each r-level set. X5 X5 X5 X5 00 i ~ i 00 i 0 i ð1 pÞ y~i ðt;rÞp ¼ CiðrÞp y~i ðt;rÞp 4 y~iðt;rÞp i¼1 i¼1 i¼!1 ! i¼1 X5 5. Numerical examples i þ4 y~iðt;rÞp þð4 4tÞ ð33Þ i¼1 Example 1. Consider the second-order fuzzy linear differential 0 equation (Xiaobin and Dequan, 2013) y~ið0; rÞ¼0; y~ið0; rÞ¼0

P 5 Table 1 Optimal values of i¼1CiðrÞ given by 5-order of OHAM for Eq. (31).

r C1ðrÞ C2ðrÞ C3ðrÞ C4ðrÞ C5ðrÞ 0 1.04098 0.00047659 0.000013094 3:5393 107 7:19469 109 0.2 1.04134 0.00049515 0.000014174 4:19711 107 2:53879 108 0.4 1.04087 0.00047063 0.000012759 3:38551 107 5:52697 109 0.6 1.04125 0.00063290 0.00040490 1:41883 107 5:33321 109 0.8 1.041203 0.00048767 0.000013721 3:82523 107 5:43775 108 1 1.041131 0.00048390 0.000013505 3:73537 107 7:88364 109

P 5 Table 2 Optimal values of i¼1CiðrÞ given by 5-order of OHAM for Eq. (31).

r C1ðrÞ C2ðrÞ C3ðrÞ C4ðrÞ C5ðrÞ 0 1.04094 0.00047412 0.000012945 3:46279 107 6:89188 109 0.2 1.04067 0.00046043 0.000012170 3:09365 107 5:60042 109 0.4 1.04035 0.00044264 0.000011113 2:56184 107 3:58411 109 0.6 1.04030 0.00043985 0.000010951 2:48296 107 3:30018 109 0.8 1.04144 0.00049992 0.000014419 4:14407 107 1:39664 108 1 1.04100 0.00047708 0.000013115 3:54585 107 7:19558 109

Table 3 Comparison of the result accuracy of 5-order OHAM at t ¼ 0:1 and the method (Xiaobin and Dequan, 2013)att ¼ 0:001 for the lower and the upper solution of Eq. (31) for all r 2½0; 1. r E r ½Errorr Xiaobin and Dequan (2013) ð0:1; Þ OHAM ½Errorr Xiaobin and Dequan (2013) Eð0:1; rÞ OHAM 0 0.00099871222831 5:77315 1015 0:00101009737569 7:99360 1015 0.2 0.00119975860278 7:11068 1010 0:00080905100122 5:15143 1014 0.4 0.00140080497724 2:75814 1011 0:00060800462676 4:97379 1014 0.6 0.00160185135171 3:11132 1010 0:00040695825230 5:68434 1014 0.8 0.00180289772617 1:41117 109 0:00020591187783 3:94784 1010 1 0.00200394410063 3:99680 1015 0:00000486550337 3:55271 1015 Optimal homotopy asymptotic method 83 where r 2½0; 1 and i = 1, 2, 3, 4, 5. Using mathematica pack- y~ 00ð0Þ¼ er; e2r age to find the solutions for the lower and the upper bounds for the problems (30) and (31), we obtain y~000ð0Þ¼½r þ 2; 4 r ; 8r 2½0; 1 ! X2 The exact solution of the Eq. (35) was given in Khodadad and ~ ~ y~ðt; rÞ¼y~0ðt; rÞþy~1ðt; C1ðrÞ; rÞþy~2 t; CiðrÞ; r Moghadam (2006). Applying OHAM in Section 3 on Eq. (35) ! i¼1 ! we obtain: X3 X4 Zero order problem þ y~3 t; C~iðrÞ; r þ y~4 t; C~iðrÞ; r i¼1 i¼1 ! y~ð4Þðt; rÞ¼0 ð36Þ X5 0 ~ þ y~5 t; CiðrÞ; r ð34Þ pffiffi r pffiffiffiffiffiffiffiffiffiffiffi i¼1 y ð0; rÞ¼ 0:3; 0:2 1 r þ 0:2 ; 0 2 By using the least square method that was mentioned in 0 r 2r y0ð0; rÞ¼ 0:4e 0:3; 0:4e 0:3 Section 3, we can compute the optimal values of C~1ðrÞ, C~ ðrÞ, C~ ðrÞ, C~ ðrÞ and C~ ðrÞ as shown in t ¼ 0:1 the following 2 3 4 5 y00ð0; rÞ¼ er; e2r ; y000ðt; rÞ¼½r þ 2; 4 r tables below (see Tables 1–3). 0 0 First to sixth order problem Now we can tabulate the absolute errors ½Er and ½Er of the approximate solutions y 0:1; r and y 0:1; r obtained by () ! ð Þ ð Þ X6 i ð4Þ 5-order OHAM series solution compared with undetermined ð1 pÞ p y~i ðt;rÞ fuzzy coefficients method in (Xiaobin and Dequan, 2013) for ( i¼1 ! ! ! all r 2½0; 1 as follows. X6 X6 X6 i ð4Þ i 000 i 00 We can conclude from the above table the accuracy of the pC1 p y~i ðt;rÞ p y~i ðt;rÞ p y~i ðt;rÞ approximate solution of Eq. (31) solved by 5-order OHAM for i¼0 ! i¼0 !) i¼0 all r 2½0; 1 when t ¼ 0:1 is better than undetermined fuzzy X6 X6 piy~0 t;r 2 piy~ t;r 37 coefficients method (Seikkala, 1987) when t ¼ 0:001. The next ið Þ ið Þ ð Þ i¼0 i¼0 figure shows the 5-order OHAM approximate solution y~ðt; rÞ compared with the exact solution y~ðt; rÞ for all r 2½0; 1 at: 0 00 000 y~ið0; rÞ¼0; y~ ð0; rÞ¼0; y~ ð0; rÞ¼0; y~ ð0; rÞ¼0 t = 0.1: (see Fig. 2). i i i where r 2½0; 1 and i =1,2,3,4,5,6. Example 2. Consider the fourth order linear FIVP with Using mathematica package to find the solutions for the triangular fuzzy initial conditions (Khodadad and lower and the upper bounds for the problems (36) and (35),we Moghadam, 2006) obtain (see Fig. 3)

ð4Þ 000 00 0 X6 y~ ðtÞ¼y~ ðtÞþy~ ðtÞþy~ ðtÞþ2y~ðtÞ; t 2½0; 1ð35Þ ~ y~ðt; rÞ¼y~0ðt; rÞþ y~1ðt; C1ðrÞ; rÞð38Þ hi pffiffi pffiffiffiffiffiffiffiffiffiffiffi i¼1 y~ð0Þ¼ 0:5 r 0:3; 0:2 1 r þ 0:2 By using the least square method that was mentioned in Section 3, we can compute the optimal values of C~ r as 0 r 2r 1ð Þ y~ ð0Þ¼ 0:4e 0:3; 0:4e 0:3 shown in the following tables below (see Table 4).

Figure 2 Approximate 5-order OHAM and the exact solution of Figure 3 Approximate 6-order OHAM and exact solutions of Eq. (31) at t = 0.1. Eq. (35) at t =1. 84 A.F. Jameel et al.

Table 4 Comparison of the result accuracy solved by 6-order of OHAM and the exact solution at t ¼ 1 for the lower and the upper solution of Eq. (35) for all r 2½0; 1.

r C1ðrÞ Eð1; rÞ OHAM C1ðrÞ C1ðrÞ 0 1.103529525572901 1:54824 108 1.103736577943245 1.103736577943245 0.2 1.103744444421023 2:11509 108 1.1037575787033083 1.1037575787033083 0.4 1.103792071040843 2:57572 108 1.1037732976880386 1.1037732976880386 0.6 1.1038080533565133 3:06465 108 1.103794588365459 1.103794588365459 0.8 1.10381293474532 3:60336 108 1.1038103704152256 1.1038103704152256 1 1.1038081037895575 4:21204 108 1.1038081037895575 1.1038081037895575

6. Conclusions Barro, S., Marin, R., 2002. Fuzzy Logic in Medicine. Physica-Verlag, Heidelberg. Bodjanova, S., 2006. Median alpha-levels of a fuzzy number. Fuzzy In this paper, we studied and applied the optimal homotopy Sets Syst. 157 (7), 879–891. asymptotic method in finding solution of high order fuzzy ini- Dubois, D., Prade, H., 1982. Towards fuzzy differential calculus, Part tial value problems involving linear ordinary differential equa- 3: differentiation. Fuzzy Sets Syst. 8, 225–233. tions. To the best of our knowledge, this is the first attempt for El Naschie, M.S., 2005. From experimental quantum optics to solving the high order FIVPs with OHAM. The method has quantum gravity via a fuzzy kahler manifold. Chaos Solitons been formulated to obtain an approximate solution of general Fract. 25, 969–977. high order FIVP. The convergence theorem of OHAM for Fard, O.S., 2009. An iterative scheme for the solution of generalized solving FIVPs has been presented and proved. In OHAM, system of linear fuzzy differential equations. World Appl. Sci. J. 7, 1597–11604. the control and adjustment of the convergence of the series Ghanbari, M., 2009. Numerical solution of fuzzy initial value solution using the convergence control parameters are problems under generalization differentiability by HPM. Int. J. achieved in a simple way in a numerical example including sec- Ind. Math. 1 (1), 19–39. ond order linear and fourth order fuzzy initial value problems Gupta, A.K., Saha Ray, S., 2014a. Comparison between homotopy showing the capability and efficiency of the OHAM. We perturbation method and optimal homotopy asymptotic method obtained accurate results by using even low order approxima- for the solution of Boussinesq–Burgers equations. Comput. Fluids tion. Moreover, this technique converges to the exact solution 103 (2014), 34–41. and requires less computational work directly without reduc- Gupta, A.K., Saha Ray, S., 2014b. 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ORIGINAL ARTICLE An efficient numerical method for computation of the number of complex zeros of real polynomials inside the open unit disk

Muhammad Mujtaba Shaikh a,*, Karem Boubaker b a Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Sindh, Pakistan b Tunis-City University Ecole Supe´rieure des Sciences et Techniques de Tunis, 63 Rue sidi Jabeur, 5100 Mahdia, Tunisia

Received 9 February 2014; revised 18 April 2015; accepted 25 April 2015 Available online 21 May 2015

KEYWORDS Abstract In this paper, a simple and efficient numerical method is proposed for computing the Real polynomials; number of complex zeros of a real polynomial lying inside the unit disk. The proposed protocol uses Complex zeros; the Boubaker polynomial expansion scheme (BPES) to build sequence of polynomials based on the Sturm sequences; concept of Sturm sequences. The method is used in a direct way without using any restrictions in Boubaker polynomials reference to other existing methods. The protocol is applied to some example polynomials of differ- ent orders and utility of the algorithm is noticed. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The Newton mapping of non-zero polynomials is also based on this notion. In fact, for a given polynomial PðzÞ, Computation of the number of non-real zeros of real polyno- the Newton mapping NPðzÞ, which is defined by: 0 mials inside the open unit disk is very important in complex NPðzÞ¼z PðzÞ=P ðzÞð3Þ analysis and system control. For example, for a corrector of the form: would be defined only if the zeros of PðzÞ are contained in the open unit disk. Xk1 Xk1 In this study, we present a new protocol for determining the u ¼ A u þ h a u0 ð1Þ nþ1 i ni i ni exact number of complex zeros of a given real polynomial in i¼0 i¼1 the unit disk using a well-known applied mathematics proto- Simpson’s stability rule is ensured if the polynomial: col, the Boubaker polynomials. The polynomials were estab-

Xk1 lished by Boubaker (2007, 2008) and have been worked upon k k1i by many researchers till now for further developments and PðzÞ¼z Aiz ð2Þ i¼0 its utilities are being investigated to deal various types of case-studies in applied engineering, medical sciences, etc. has all of its zeros in the open unit disk. Several properties and modified versions of these polynomi- * Corresponding author. Cell: +92 333 2617602. als have been investigated; to mention a few studies: Boubaker E-mail address: [email protected] (M.M. Shaikh). et al. (2007), Labiadh (2007), Oyodum et al. (2009), Zhao et al. Peer review under responsibility of University of Bahrain. (2009, 2010) and Barry and Hennessy (2010). A modified http://dx.doi.org/10.1016/j.jaubas.2015.04.005 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Numerical method for computation of the number of complex zeros of real polynomials 87 version of these polynomials, called 4q-Boubaker polynomials, This follows the procedure of constructing the Sturm-like was the basis for the development of the Boubaker polynomi- shaped sequence of the polynomials. In Section 3, we use the als expansion scheme (BPES). The scheme has been used by protocol to determine the exact number of complex zeros of Agida and Kumar (2010) and Kumar (2010) to solve particular some variable degree polynomials in the open unit disk. We integral equations. On the other hand few standard boundary end with illustrating conclusion and future work. value problems of ordinary differential equations (Boubaker, 2008; Zhang and Naing, 2010; Koc¸ak et al., 2011a) and many 2. Materials and methods physical models involving ordinary differential equations sys- tems (Milgram, 2011; Dubey et al., 2010, Yildirim et al., 2.1. The Boubaker polynomials 2010) were solved more efficiently by BPES as compared to other methods. Physical models in terms of partial differential The first monomial definition of the Boubaker polynomials equations in many fields were reliably addressed through (Boubaker, 2007, 2008; Ghanouchi and Labiadh, 2008; BPES. For example: the works carried out by Ghrib et al. Belhadj et al., 2009) appeared in a physical study that yielded (2008), Guezmir et al. (2009) and Zhang and Li (2010) in an analytical solution to the heat equation inside a physical general to investigate material and alloy properties and more model. particularly the works by Zhang (2010b) and Slama et al. (2008a,b, 2009, 2010) in the field of resistance spot welding Definition 1. Boubaker polynomials monomial definition is research to obtain analytical temperature distribution. given by: The contributions by Ghanouchi and Labiadh (2008), Tabatabaei et al. (2009), Belhadj et al. (2009) and Koc¸ak XnðnÞ ðn 4pÞ et al. (2011b) further evoked the use of BPES to solve core B ðXÞ¼ Cp ð1Þp Xn2p ð4Þ n n p np studies in the field of Heat and Mass Transfer. Awojoyogbe p¼0 ð Þ and Boubaker (2009) and in many other studies jointly where: explained how NMR blood flow equations can be solved in jk n various heart models to find magnetic phase shift, and in n 2n þ ðð1Þ 1Þ nðnÞ¼ ¼ Bio-medical engineering to find net magnetization under the 2 4 MRI exposure in various geometries. The work carried out (The symbol: bc designates the floor function). by Fridjine and Amlouk (2009) discusses the case of optimiz- ing functional materials in hybrid solar energy devices. The Boubaker polynomials have also the explicit monic The main idea in this paper consists of constructing the expression: "# Sturm-sequences which are built using the properties of XnðnÞ 2Yp1 ðn 4pÞ BPES. The idea of this construction is based on the work of B ðXÞ¼Xn ðn 4Þ:Xn2 þ ðn jÞ ð1Þp Xn2p n p! Schelin (1983) who first used Chebyshev polynomials to con- p¼2 j¼pþ1 struct Sturm-like sequence to count zeros of real polynomials. ð5Þ A similar construction using Chebyshev polynomials appears in the works of Locher and Skrzipek (1995) and Gleyse (1997). The examination of the number of sign changes and Theorem 1. The characteristic recurrence relation for the the sign repetitions in the built-off Sturm sequences in this Boubaker polynomials is: work using 4q-Boubaker polynomials finally leads to define the complete protocol to achieve the goal of computing the B ðXÞ¼X B ðXÞB ðXÞ for : m > 2 number of complex zeros of real polynomials. m m1 m2 The concept of sign changes and sign repetitions dates back P hi nðm1Þ ðm14pÞ p to Seventeenth century when Rene Descarte proposed a rule of Proof. For m >2: Bm1ðXÞ¼ C p¼0 hiðm1pÞ m1p signs to find upper bound on the count of positive and negative P ð1Þp Xm12p, and: B ðXÞ¼ nðm2Þ ðm24pÞ Cp real zeros of a polynomial. Another concept of examining signs m2 p¼0 ðm2pÞ m2p p appears in Routh–Hurwitz test and its extensions ð1Þ Xm22p. By calculating the amount: (Gantmacher, 1960) which is used to determine if all zeros of D ¼ X Bm1ðXÞBm2ðXÞ, it gives: a real polynomial lie in the open left-half plane and hence to " nðXm1Þ comment on polynomial stability. However, the criterion of ðm 1 4pÞ D ¼ Xm Cp ð1Þp X2p counting the number of sign changes and the sign repetitions m 1 p m1p p¼0 ð Þ used to develop method in this paper is based on a similar con- # nðXm2Þ cept used in Sturm theorem (Collins and Rudiger, 1983)to ðm 2 4pÞ Cp ð1Þp X22p count real zeros of polynomials defined in interval [1,1]. m 2 p m2p p¼0 ð Þ We demonstrate through worked out examples in Section 3 that the proposed protocol – which uses little extension of which gives: the concepts in the Sturm theorem – yields encouraging results D ¼ X Bm1ðXÞBm2ðXÞ when it comes to count the number of complex zeros of real XnðnÞ polynomials in the open unit disk. ðn 4pÞ ¼ Xm Cp ð1Þp X2p ¼ B ðXÞ The structure of this paper is as follows: n p np m p¼0 ð Þ We begin by introducing, in Section 2, some necessary definitions and mathematical preliminaries of the Boubaker The ordinary generating function of the Boubaker polyno- Polynomials which are required for establishing our results. mials is: 88 M.M. Shaikh, K. Boubaker

1 þ 3t2 2.2. Built-off Sturm shaped sequence f ~ðX; tÞ¼ ð6Þ B 1 þ tðt XÞ Zhao et al. (2010) investigated some special properties of Definition 2. A Sturm shaped sequence of polynomials is a set: the Boubaker polynomials Bn for the case n =4q which include involvement of only even powers of x in the polynomi- fP0ðxÞ; P1ðxÞ; P2ðxÞ; ...; PMðxÞg ð10Þ als and removal of the 2q rank monomial terms from the expli- with P ; P and P three initializing nonzero polynomials, M a cit form. In particular, these properties lead to explicit 0 1 2 given integer and P j verifying: expressions with only 2q effective terms and hence to a class i i¼1...M of polynomials which are all even functions. Correspondent PiðxÞ¼UiðxÞPi1ðxÞþPi2ðxÞ; i P 2 ð11Þ 4q-order Boubaker polynomials (Zhao et al., 2010) are pre- where UiðxÞji¼2...N is a given polynomial sequence. sented in Eq. (7) as a general form and Eq. (8) as first P functions: N i Let us consider a real polynomial QðxÞ¼ i¼0nix , along with the sequence fP ðxÞ; P ðxÞ; P ðxÞ; ...; P ðxÞg: X2q 0 1 2 N ðq pÞ 8 B ðXÞ¼4 Cp ð1Þp X2ð2qpÞ ð7Þ > Xn 4q ð4q pÞ 4qp > p¼0 > P0ðxÞ¼ niB4iðxÞ > > i¼0 8 > Xn > B X > > 0ð Þ¼1; > P x n B x > < 1ð Þ¼ i 4i4ð Þ > 4 i¼1 > B4ðXÞ¼X 2; 12 > > ð Þ > > P2ðxÞ¼B4ðxÞP1ðxÞP0ðxÞ > B X X8 4X6 8X2 2; > > 8ð Þ¼ þ > > > ... <> 12 10 8 4 2 > B12ðXÞ¼X 8X þ 18X 35X þ 24X 2; > > Pkþ1ðxÞ¼B4ðxÞPkðxÞPk1ðxÞ > 16 14 12 10 6 4 2 : > B ðXÞ¼X 12X þ 52X 88X þ 168X 168X þ 48X 2; > 16 P ðxÞ¼B ðxÞP ðxÞP ðxÞ > N 4 N1 N2 > 20 18 16 14 12 8 6 > B20ðXÞ¼X 16X þ 102X 320X þ 455X 858X þ 1056X > Here N is order of the real polynomial Q(x) and n is number > > 495X4 þ 80X2 2; of non-zero terms in Q(x). > :> ... ~ ð8Þ Theorem 3. The sequence PNðxÞ¼fP0ðxÞ; P1ðxÞ; P2ðxÞ; ...; PNðxÞg with polynomials as in (12) is a Sturm shaped sequence The proposed protocol in this paper is based on 4q-order constructed from Boubaker polynomials. Boubaker polynomials instead of original polynomials Bn due to the benefits that all 4q-order polynomial are even func- Proof. We have, for all values of 0 6 k < M: tions and result in less computational cost (to be elaborated in Section 3). We quote the following important results of 4q- Pk1ðxÞ¼B4ðxÞPkðxÞPkþ1ðxÞ¼KPkðxÞþr; Boubaker polynomials (Zhao et al., 2010) which will be useful in the construction of the Sturm shaped sequences and the final the remainder r of the Euclidian division of Pk1ðxÞ by PkðxÞ is h implementation of the protocol to follow. Readers can refer to hence: r ¼Pkþ1ðxÞ. (Zhao et al., 2010) for detailed proofs. Proposed protocol. For a sequence P~NðxÞ¼fP0ðxÞ; P1ðxÞ; Theorem 2. The following equality holds: P2ðxÞ; ...; PNðxÞg, associated to a polynomial QðxÞ¼ N i i¼0nix , defined in the domain ½1; 1, the number Z1;Q of Xn complex zeros inside the unit disk is given by: BkðxÞBkðyÞ k¼0 Z ¼ Sð1ÞþSð1Þð13Þ B ðxÞB ðyÞB ðxÞB ðyÞ 1;Q ¼ 3 þ nþ1 n n nþ1 for all x–y x y where SðxÞ¼SCðxÞSRðxÞ represents the difference between the number of sign changes and sign repetitions in

Proof. As a consequence of recurrence relation (Theorem 1) the sequence P~NðxÞ. and assuming: This protocol is an extension of the Sturm theorem for real D D zeros of real-coefficient polynomials. For a proof, refer to B ðxÞB ðyÞ¼ k k1 for k ¼ 2; 3; ... ð8Þ k k x y Collins and Rudiger (1983). While the usual Sturm theorem and related works on Sturm-like sequence using Chebyshev where: Dk ¼ Bkþ1ðxÞBkðyÞBkðxÞBkþ1ðyÞ (8) summed from 0 polynomials in literature (Schelin, 1983; Locher and to n gives the desired formula. h Skrzipek, 1995; Gleyse, 1997) target only the number of real zeros of real coefficient polynomials in open unit disk or other If x ! y in (8), we obtain the following Corollary. annulus, we demonstrate through examples in the next section that the proposed protocol – an extension to the theorem – can Corollary 1. The following equality is satisfied be used to count number of complex zeros of real polynomials

Xn in open unit disk. 2 0 0 ~ BkðxÞ¼3 þ Bnþ1ðxÞBnðxÞBnðxÞBnþ1ðxÞ; n > 0 ð9Þ Since the built-off Sturm sequence PNðxÞ is constructed k¼0 through 4q-Boubaker polynomials, which are all even, as a Numerical method for computation of the number of complex zeros of real polynomials 89 consequence the number of sign changes and also the sign Q1ðxÞ gives the following Boubaker polynomial built Sturm- repetitions at 1 and 1 will be the same, i.e. SCð1Þ¼ shaped sequence: C R R 8 S ð1Þ and S ð1Þ¼S ð1Þ)S ð1Þ¼S ð1Þ. Thus, (13) can Xn > equivalently be expressed as: > 9 281 >P0ðxÞ¼ niB4iðxÞ¼B16ðxÞþ8 B12ðxÞþ256B8ðxÞ > > i¼0 Z1;Q ¼ 2S ð1Þ¼2S ð1Þð14Þ < 225 625 512 B4ðxÞþ4096B0ðxÞ It can be noted that the use of 4q-Boubaker polynomials > Xn > 9 281 225 minimizes the computational cost of (13) by half as one needs >P1ðxÞ¼ niB4i4ðxÞ¼B12ðxÞþ8B8ðxÞþ256 B4ðxÞ512 B0ðxÞ > to count the sign changes and repetitions either only at 1 or 1 :> i¼1 as in (14). Pkþ1ðxÞ¼B4ðxÞPkðxÞPk1ðxÞ; k ¼ 2;3;...;8

3. Results and discussion and corresponding sign sequence fþ; ; ; þ; ; ; þ; ; g at x =1orx = 1. Consequently:

The described protocol has been applied on following polyno- Z1;Q1 ¼ 2S ð1Þ¼2S ð1Þ¼2ð5 3Þ¼4 mials (all zeros are shown opposite): Example 1: which is true as only four complex zeros of Q1ðxÞ: 1 1 1 1 2 þ i; 2 i; 2 þ i and 2 i lie in the open unit disk. Zeros 8 9 6 281 4 225 2 Q1ðxÞ¼x þ x þ x x loci for Q1ðxÞ are shown in Fig. 1. 8 256 512 625 1 1 1 1 1 1 þ i; i; i; i 1.5 4096 2 2 2 4 2 4 Example 2: 1 5 4 15 3 35 2 Q2ðxÞ¼x þ 4x þ x þ x þ 14x 2 2 1 1 0.5 þ 6 3; 2i; i 2 2 Example 3: 0 1 1 1 7 5 3 ffiffiffi Q3ðxÞ¼x þ x x x 0; 1; i; p i 2 2 2 Imaginary Part -0.5 Example 4: 1 1 1 Q ðxÞ¼x3 x2 þ x 1; i 4 4 4 2 -1 Implementation details of the proposed protocol on polynomials in Examples 1–4 are given in Table 1 with specific -1.5 values, sign sequence and sign patterns in corresponding -1.5 -1 -0.5 0 0.5 1 1.5 Sturm-shaped sequences. We explicitly describe implementa- Real Part tion on, say, Q1ðxÞ. The application of the protocol on Fig. 1 Zeros loci for Q1ðxÞ.

Table 1 Implementation details of the proposed protocol on example polynomials 1–4.

Q1ðxÞ Q2ðxÞ Q3ðxÞ Q4ðxÞ

P0 341/537 57/2 0 11/2 P1 83/512 10 3/2 17/4 P2 1143/2414 77/2 3/2 5/4 P3 341/537 57/2 0 11/2 P4 83/512 10 3/2 – P5 1143/2414 77/2 3/2 – P6 341/537 – 0 – P7 83/512 – 3/2 – P8 1143/2414 – – – Sign sequence at x =1or1 fþ; ; ; þ; ; ; þ; ; g fþ; þ; ; þ; þ; g fþ; þ; ; þ; þ; ; þ; þg f; þ; þ; g SCð1Þ or SCð1Þ 5342 SRð1Þ or SRð1Þ 3231 Sð1Þ or Sð1Þ 2111 Z1;Q 4222 Bold values represent main output of the proposed protocol on examples 1–4. These bold values represent the number of complex zeros inside the unit disk, found by the protocol for polynomials in examples 1–4. 90 M.M. Shaikh, K. Boubaker

2.5 1.5

2 1 1.5

1 0.5 0.5

0 0

-0.5 Imaginary Part Imaginary Part -0.5 -1

-1.5 -1

-2

-1.5 -2.5 -1.5 -1 -0.5 0 0.5 1 1.5 -3 -2 -1 0 1 Real Part Real Part

Fig. 4 Zeros loci for Q4ðxÞ. Fig. 2 Zeros loci for Q2ðxÞ.

1.5 using the Boubaker polynomial generated Sturm sequence. The protocol is general and efficient since no restriction is applied to the targeted polynomial. According to the example 1 investigations, this method is a simple and efficient numerical method for computing the number of polynomials complex zeros lying inside the unit disk. 0.5 The construction of Boubaker polynomials built-off Sturm shaped sequences in this work to exactly compute the number of complex zeros for real polynomials, as a start, through this 0 work further encourages researchers to revisit the approxima- tions, work to refine it for other similar problems and devise

Imaginary Part extended methods. Investigation of further accuracy and suit- -0.5 ability of this protocol along with proposition of its utility to address case studies in complex analysis and control theory are the topics of future research. It can be observed that no -1 conditions are presumed for the polynomial QðxÞ, as opposed to the methods (Gleyse, 1997; Gleyse and Larabi, 2011). Comparison of our method will be made with those of the -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Cauchy-indices-related method, used by Gleyse (1997),or Real Part those of methods using Schur-Cohn, Brown and Cohn trans- forms (Gleyse and Larabi, 2011) from the view-point of anal-

Fig. 3 Zeros loci for Q3ðxÞ. ysis of order of complexity in future.

The results in Table 1 speak for themselves. The proposed Acknowledgement protocol shows that polynomials in Examples 1–4 have 4, 2, 2, and 2 complex zeros inside the open unit disk, respectively, Authors are thankful to their institutes for providing facilities which is in good agreement with the loci plots of these polyno- for conducting this research. Authors are also highly indebted mials in z-plane (Figs. 1–4). It can be noted that the proposed to referees for highlighting important gaps and key points to method counts only those complex zeros inside open unit disk adequately improve accuracy in the present version of our work. that are purely non-real, i.e. involve some imaginary term. This is evident through Example-3 and meanwhile from Fig. 3 that References x = 0 (a purely real zero of Q3ðxÞ) – beside located in the open unit disk – is not counted by the proposed algorithm. Agida, M., Kumar, A.S., 2010. A Boubaker polynomials expansion scheme solution to random Love’s equation in the case of a rational 4. Conclusion and future work Kernel. Electron. J. Theor. Phys. 7 (24), 319–326. Awojoyogbe, O.B., Boubaker, K., 2009. A solution to Bloch NMR flow equations for the analysis of hemodynamic functions of blood The exact number of complex zeros in the open unit disk of flow system using m-Boubaker polynomials. Current Appl. Phys. 9 some polynomials of different orders has been determined (1), 278–283. Numerical method for computation of the number of complex zeros of real polynomials 91

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(2), 141–146. Dubey, B., Zhao, T.G., Jonsson, M., Rahmanov, H., 2010. A solution Slama, S., Bessrour, J., Boubaker, K., Bouhafs, M., 2008b. A to the accelerated-predator-satiety Lotka–Volterra predator–prey dynamical model for investigation of A3 point maximal spatial problem using Boubaker polynomial expansion scheme. J. Theor. evolution during resistance spot welding using Boubaker polyno- Biol. 264 (1), 154–160. mials. Eur. Phys. J. Appl. Phys. 44 (03), 317–322. Fridjine, S., Amlouk, M., 2009. A new parameter: an abacus for Slama, S., Bessrour, J., Bouhafs, M., Mahmoud, K.B., 2009. optimizing pv–t hybrid solar device functional materials using the Numerical distribution of temperature as a guide to investigation Boubaker polynomials expansion scheme. Mod. Phys. Lett. B 23 of melting point maximal front spatial evolution during resistance (17), 2179–2191. spot welding using Boubaker polynomials. Numer. Heat Transfer Gantmacher, Felix R., 1960. In: Theory of Matrices, vol. 2. 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ORIGINAL ARTICLE Real fixed points and dynamics of one parameter x family of function ðb 1Þ=x

Mohammad Sajid

College of Engineering, Qassim University, P.O. Box: 6677, Buraidah, Al-Qassim, Saudi Arabia

Received 21 June 2015; revised 18 September 2015; accepted 11 October 2015 Available online 2 November 2015

KEYWORDS Abstract In this paper, the real fixed points and dynamics of one parameter family of functions k ; k > x = ; – > ; – Bifurcation; fkðxÞ¼ fðxÞ 0, where fðxÞ¼ðb 1Þ x x 0 and fð0Þ¼ln b for b 0 b 1, are investi- < < Dynamics; gated. The real fixed points of fkðxÞ as well as their nature are explored. For 0 b 1, it is seen Fixed points that one fixed point of fkðxÞ is attracting and one fixed point is repelling for 0 < k < k and fkðxÞ has no real fixed points for k > k. It is also found that the bifurcation in the real dynamics of fkðxÞ occurs at the real parameter value k . For b > 1, similar results are shown. Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction (2012), and Sajid and Kapoor (2007). Such investigations are interesting for the description of Julia sets, Fatou sets and The dynamics of one parameter family of exponential map has other studies in complex dynamics. In the present paper, the x = been vastly studied by Devaney (2001), Devaney (2004), and real dynamics of one parameter family of function ðb 1Þ x > ; – Shen and Rempe-Gillen (2015). The investigation of the for b 0 b 1 is explored which is a generalization of one x = dynamical properties of this exponential family is simpler than parameter family of function ðe 1Þ x by Kapoor and x = other kinds of families which include exponential map, Prasad (1998). The real dynamics of the function ðb 1Þ x, > (Fagella and Garijo, 2003; Kuroda and Jang, 1997; Petek for b 1, is similar to the dynamics of the function x = < < and Rugelj, 1998; Yanagihara and Gotoh, 1998). The real ðe 1Þ x and for 0 b 1, is somewhat similar to the x = > dynamics of the cubic polynomials, generalized logistic maps dynamics of ðe 1Þ x. To achieve this goal, for b 0 and b – 1, the following function is assumed: and one parameter family of transcendental functions was ( found in Akbari and Rabii (2015); Magrenan and Gutierrez b x1 if x – 0 (2015); Radwan (2013); and Sajid and Alsuwaiyan (2014) fðxÞ¼ x respectively. The real dynamics of functions has become an ln b if x ¼ 0 important research area, partially due to the dynamics in the Consider one parameter family of function fðxÞ complex plane which was induced using the real dynamics by k : k > Kapoor and Prasad (1998), Nayak and Prasad (2014), Sajid G¼fgfkðxÞ¼ fðxÞ 0 A point x is said to be a fixed point of function fðxÞ if Tel.: +966 507017848. fðxÞ¼x. A fixed point xf is called attracting, neutral (indiffer- 0 < ; 0 0 > E-mail addresses: [email protected], [email protected]. ent) or repelling if jf ðxfÞj 1 jf ðxfÞj ¼ 1orjf ðxfÞj 1 Peer review under responsibility of University of Bahrain. respectively. http://dx.doi.org/10.1016/j.jaubas.2015.10.001 1815-3852 Ó 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Real fixed points and dynamics of one parameter family of function ðb x 1Þ=x 93

In Theorems 2.1 and 2.2, the real fixed points of fk 2Gand Remark 2.2. It is observed that k ¼ k . the nature of these fixed points are found for 0 < b < 1 and The following theorems describe the real fixed points of b > 1 respectively. The real dynamics of fk 2G is shown in fk 2Gand their nature for 0 < b < 1 and b > 1 respectively: Theorems 3.1 and 3.2 for 0 < b < 1 and b > 1 respectively.

Theorem 2.1. Let fk 2Gand 0 < b < 1. 2. Real fixed points of fk ‰ G and their nature (a) If 0 < k < k , then f kðxÞ has only two real fixed points, r1 Let f x b x1 ; x – 0 and f 0 ln b. Further, suppose that ð Þ¼ x ð Þ¼ and a1 such that r1 < a1, where r1 is repelling and a1 is attracting. 0 1 x /ðxÞ¼fðxÞxf ðxÞ¼ ½ð2 x ln bÞb 2 k k f x x x (b) If ¼ , then kð Þ has only one real fixed point 1 and that is rationally indifferent. For 0 < b < 1, the function fðxÞ is strictly increasing nega- (c) If k > k , then f kðxÞ has no real fixed points. tive valued function and f 0ðxÞ is strictly decreasing positive val- ued functions. Then, the function /ðxÞ is strictly decreasing in ; ; the interval ð1 0 and is negative in the interval ½0 þ1Þ. R / < / Proof. Suppose hkðxÞ¼fkðxÞx for x 2 . Then, it is seen Moreover, ð0Þ¼ln b 0 and ðxÞ!þ1 as x !1. R ; that (i) hkðxÞ is continuously differentiable in and negative Therefore, there exists x1 2 ð1 0Þ such that 8 for jxj sufficiently large since hkðxÞ!1 as x !1 and > > < < 0 for x x1 x !1. (ii) hkðxÞ has a unique local maximum at / x~ ð¼ x~ ðkÞÞ since h0 ðxÞ is decreasing, h0 ðxÞ!1as ðxÞ ¼ 0 for x ¼ x1 ð1Þ 1 1 k k :> 0 < 0 for x < x < 1 x !þ1and hkðxÞ!þ1as x !1. Hence, there exists a 1 0 unique real number x~1 such that hkðx~1Þ > 0 for where x is the unique negative real root of the equation 0 0 1 x < x~1; hkðx~1Þ¼0 and hkðx~1Þ < 0 for x > x~1. ð2 x ln bÞb x 2 ¼ 0. > 0 Similarly, for b 1, the functions fðxÞ and f ðxÞ are strictly (a) Since h0 ð~x Þ¼0, it gives that k ¼ 1=f 0ð~x Þ.By / k 1 1 increasing positive valued functions. Hence, ðxÞ is strictly 0 < k < k; f 0ðxÞ < f 0ð~x Þ. It follows that x > ~x ; ; 1 1 1 1 decreasing in ½0 þ1Þ and is positive in ð1 0. Moreover, since f 0ðxÞ is decreasing. Therefore, by Eq. (1), /ð0Þ¼ln b > 0 and /ðxÞ!1as x !þ1. Consequently, 0 0 /ð~x Þ¼f ð~x Þ~x f ð~x Þ¼f ð~x Þhkð~x Þ > 0. So, ; 1 1 1 1 1 1 there exists x2 2ð0 1Þ such that h ~x > h ~x > 8 kð 1Þ 0. Using (i) and (ii) with kð 1Þ 0, the > > < < hk x f x < 0 for 1 x x2 function ð Þ has only two zeros. Hence, kð Þ r a /ðxÞ ¼ 0 for x ¼ x ð2Þ has only two real fixed points 1 and 1 with :> 2 r < ~x < a h0 r > h0 a < < > 1 1 1. Moreover, kð 1Þ 0 and kð 1Þ 0 0 for x x2 0 since r1 < ~x1 < a1. It gives that f kðr1Þ > 1 and f 0 a < r where x2 is the unique positive real root of the equation kð 1Þ 1. Thus, the point 1 is repelling and the x ð2 x ln bÞb 2 ¼ 0. point a1 is attracting for 0 < k < k . k k ~x x h x (b) If ¼ , then 1 ¼ 1 and kð 1Þ¼0. It follows that h x f x Remark 2.1. The relationship between x1 and x2 is found as kð Þ has only one zero. Hence, kð Þ has only one k k h0 x x2 ¼x1. It represents that x2 is a reflection of x1 about y-axis real fixed point for ¼ . Since kð 1Þ¼0, it shows f 0 x x which can be easily seen by Fig. 1. that kð 1Þ¼1. Consequently, the point 1 is a rationally indifferent for k ¼ k. Let us define (c) Using similar arguments as part (a), if k > k, then ~x > x and hk ~x < 0. By (ii), hk x < hk ~x < 0 1 1 1 1 ð 1Þ ð Þ ð 1Þ k k x R h x ¼ 0 and ¼ 0 for all 2 . Hence, kð Þ has no zeros. Thus, the f ðx1Þ f ðx2Þ function f kðxÞ has no real fixed points for k > k .

This completes the proof of theorem for 0 < b < 1. h

Theorem 2.2. Let fk 2Gand b > 1.

(i) If 0 < k < k , then f kðxÞ has only two real fixed points, r2 and a2 such that a2 < r2, where r2 is repelling and a2 is attracting. k k f x x (ii) If ¼ , then kð Þ has only one real fixed point 2 and that is rationally indifferent. (iii) If k > k , then f kðxÞ has no real fixed points.

Proof. Let gkðxÞ¼fkðxÞx for x 2 R. Similarly as Theo- rem 2.1, (I) gkðxÞ is continuously differentiable in R and posi- x 1 tive for jxj sufficiently large since gkðxÞ!þ1 as x !1 Fig. 1 Graphs of function y ¼ð2 x ln bÞb 2 for b ¼ 2 and b ¼ 2. and x !1. (II) gkðxÞ has a unique local minimum at 94 M. Sajid

0 0 x~2ð¼ x~2ðkÞÞ since gkðxÞ is strictly increasing, gkðxÞ!1as The proof of theorem is completed for 0 < b < 1. h 0 x !1and gkðxÞ!þ1as x !þ1. Therefore, there exists 0 a unique real number x~2 such that gkðx~2Þ < 0 for Theorem 3.2. Let fk 2Gand b > 1. 0 0 x < x~2; gkðx~2Þ¼0 and gkðx~2Þ > 0 for x > x~2. n (i) If 0 < k < k , then f kðxÞ!a2 as n !1for x < r2 and The rest proof of the theorem is similar as Theorem 2.1. n f kðxÞ!1as n !1for x > r2. Moreover, for b ¼ e > 1, the proof of this theorem can be n (ii) If k ¼ k, then f ðxÞ!x as n !1 for x < x and deduced from Kapoor and Prasad (1998). h k 2 2 f n x n x > x kð Þ!1as !1for 2. n (iii) If k > k , then f kðxÞ!1as n !1for all x 2 R. Remark 2.3. The function fk 2Ghas no real periodic points of period greater than one. It is concluded as: For 0 < b < 1, since fkðxÞ < 0 for all x 2 R, then any real periodic point of 0 ðx ln b1Þb xþ1 Proof. The function f x k is strictly increasing, fkðxÞ is negative. Moreover, fkðxÞ is strictly increasing in R , k ð Þ¼ x2 0 0 it gives that this point must be a fixed point. Similarly, for fk ðxÞ!0asx !1and fk ðxÞ!1as x !1. Then, there 0 b > 1, since fkðxÞ > 0 for all x 2 R, then any real periodic exists a unique real number x~2 such that fk ðxÞ < 1 for 0 0 point of fkðxÞ is positive. Further, fkðxÞ is strictly increasing x < x~2; fk ðx~2Þ¼1 and fk ðxÞ > 1 for x > x~2. Therefore, the þ in R , hence this point must be a fixed point. function fkðxÞx attains a local minimum value at x ¼ x~2 00 since fk ðx~2Þ > 0.

3. Real dynamics of fk ‰ G The rest of the proof of the theorem is similar to Theorem 3.1. Moreover, the proof of this theorem can be also deduced from Kapoor and Prasad (1998) for b ¼ e > 1. In the following theorems, the real dynamics of fk 2Gis found h for 0 < b < 1 and b > 1 respectively:

Remark 3.1. By Remark 2.2, the dynamics of fk 2G can be Theorem 3.1. Let fk 2Gand 0 < b < 1. directly deduced from Theorem 3.1 for b > 1, and it is also n observed that the dynamical behavior is symmetrical. (a) If 0 < k < k , then f kðxÞ!a1 as n !1for r1 < x and f n x n x < r kð Þ!1as !1for 1. For 0 < b < 1(b > 1), from Theorem 3.1 (Theorem 3.2), it k k f n x x n x < x (b) If ¼ , then kð Þ! 1 as !1 for 1 and signifies that bifurcation in the dynamics of fk 2Goccurs at the n f x n x < x kð Þ!1as !1for 1. parameter value k ¼ k (k ¼ k ) since under iteration of fk the k > k f n x n x R (c) If , then kð Þ!1as !1for all 2 . orbits of all the points greater than rk (less than rk) for < k < k < k < k k k k k 0 (0 ) and x1 (x2) for ¼ ( ¼ ) remain bounded and the orbits of all the points less than 0 k ðx ln b1Þb xþ1 (greater than) these points become unbounded; while, if Proof. The function fk ðxÞ¼ x2 is strictly decreasing, 0 0 k > k (k > k), there is no real point whose orbit remain fk ðxÞ!0asx !þ1and fk ðxÞ!þ1as x !1. Hence, 0 bounded. there exists a unique real number x~1 such that fk ðxÞ > 1 for < ~ ; 0 ~ 0 < > ~ x x1 fk ðx1Þ¼1 and fk ðxÞ 1 for x x1. Therefore, the 4. Conclusions function fkðxÞx attains a local maximum value at x ¼ x~1 since f 00ðx~ Þ < 0. k 1 The fixed points and real dynamics of one parameter family of b x1 functions fkðxÞ¼k have been investigated in this work. It (a) By Theorem 2.1(a), for 0 < k < k , it is seen that f kðxÞ x has been found that for certain range of parameter values, the has a repelling fixed point r1 and an attractive fixed real fixed points of fkðxÞ exist but for other values there are no point a1 with r1 < a1. It is easily shown that fixed points. The bifurcation in the real dynamics of fkðxÞ has f kðxÞx < 0inð1; r1Þ[ða1; 1Þ and f kðxÞx > 0 also occurred at the real parameter value. in ðr1; a1Þ. Since f kðxÞ is increasing in R , the sequence f n x is increasing and bounded above by a for f kð Þgn>0 1 Acknowledgment r1 < x < a1 and, is decreasing and bounded below by a a < x 1 for 1 . Hence, by monotone convergence theo- The author is grateful to the anonymous referee for carefully f n x a n r < x rem, kð Þ! 1 as !1 for 1 . Further, if reviewing the manuscript and giving helpful suggestions and x < r f n x n f x < x 1, kð Þ!1 as !1 since kð Þ for comments. x < r1. (b) If k ¼ k ,byTheorem 2.1(b), f kðxÞ has a unique ration- References x x f x ally indifferent fixed point at ¼ 1. Since kð Þ is n increasing in R , the sequence ff kðxÞg is decreasing Akbari, M., Rabii, M., 2015. Real cubic polynomials with a fixed point x x < x of multiplicity two. Indagationes Mathematicae 26, 64–74. http:// and bounded below by 1 for 1 . Therefore, f n x x n x < x x < x dx.doi.org/10.1016/j.indag.2014.06.001. k ð Þ! 1 as !1 for 1 .If 1, then n n Devaney, R.L., 2001. Se x: dynamics, topology, and bifurcations of f kðxÞ!1 as n !1 since the sequence ff kðxÞg is complex exponentials. Topology Appl. 110, 133–161. http://dx.doi. decreasing and unbounded below. f x < x R; f x < x x < org/10.1016/S0166-8641(00)00099-7. (c) Since kð Þ 0 for all 2 kð Þ for 0 and, Devaney, R.L., 2004. A survey of exponential dynamics. In: Aulbach, by Theorem 2.1(c), it can be deduced that, for B., Elaydi, S., Ladas, G. (Eds.), New Progress in Difference k > k; f n x n x R kð Þ!1as !1for all 2 . Equations. Chapman and Hall/CRC’, pp. 105–122. Real fixed points and dynamics of one parameter family of function ðb x 1Þ=x 95

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ORIGINAL ARTICLE Adjusted ridge estimator and comparison with Kibria’s method in linear regression

A.V. Dorugade

Y.C. Mahavidyalaya, Halkarni, Tal-Chandgad, Kolhapur, Maharashtra- 416552, India

Received 12 November 2014; revised 6 April 2015; accepted 25 April 2015 Available online 21 May 2015

KEYWORDS Abstract This paper proposes an adjusted ridge regression estimator for b for the linear regression Ridge regression; model. The merit of the proposed estimator is that it does not require estimating the ridge param- Ridge estimator; eter k unlike other existing estimators. We compared our estimator with an ordinary least squares Mean square error; (LS) estimator and with some well known estimators proposed by Hoerl and Kennard (1970), ordi- Simulation nary ridge regression (RR) estimator and generalized ridge regression (GR) and some estimators proposed by Kibria (2003) among others. A simulation study has been conducted and compared for the performance of the estimators in the sense of smaller mean square error (MSE). It appears that the proposed estimator is promising and can be recommended to the practitioners. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction (1970). Ridge regression approach has been studied by McDonald and Galarneau (1975), Swindel (1976), Lawless Regression analysis is one of the frequently used tools for fore- (1978), Singh and Chaubey (1987), Sarkar (1992), Saleh and casting in almost all disciplines; hence estimation of unknown Kibria (1993), Kibria (2003), Khalaf and Shukur (2005), parameters is a common interest for many users. These esti- Zhong and Yang (2007), Batah et al. (2008), Yan (2008), mates can be found by various estimation methods. The easiest Yan and Zhao (2009), Muniz and Kibria (2009), Yang and and the most common method of them is the ordinary least Chang (2010), Khalaf (2012) and Dorugade (2014) and others. squares (LS) technique, which minimizes the squared distance Ridge Regression estimator has been the benchmarked for between the estimated and observed values. Multicollinearity almost all the estimators developed later in this context. among the explanatory variables in the regression model is Most of the researchers compare superiority of their suggested an important problem that exhibits serious undesirable effects estimators with LS, RR, GR and other existing methods in on the analysis faced in applications. The LS estimator is sen- terms of minimum MSE criterion in the presence of multi- sitive to number ‘errors’, namely, there is an ‘explosion’ of the collinearity. In this article, our primary aim is to suggest an sampling variance of the estimators. Alternative estimators are estimator by modifying the ordinary ridge regression (RR) designed to combat multicollinearity-yield-biased estimators. estimator avoiding the computation of ridge parameter and One of the popular numerical techniques to deal with mul- secondly to evaluate the performance of our estimator with ticollinearity is the ridge regression due to Hoerl and Kennard LS, RR and GR estimators in the presence of sever or extre- mely sever multicollinearity. E-mail address: [email protected] This article is organized as follows: in Section 2, we define model and parameter estimation methods with their bias and Peer review under responsibility of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2015.04.002 1815-3852 ª 2015 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Adjusted ridge estimator and comparison with Kibria’s method 97

Figure 3 ‘‘fm’’ for AR, RR and GR estimators (q = 0.999, p = 3 and b = (14, 5, 2, 6)0). Figure 1 ‘‘fm’’ for AR, RR and GR estimators (q = 0.95, p =3 and b = (10, 4, 1, 8)0). 0 0 0 eigenvalue of X X and T T ¼ TT ¼ Ip. We obtain the equiva- lent model MSE. In Section 3, we have proposed biased estimator. We Y ¼ Zr þ e; ð2Þ compare our new estimator in the MSE sense, with the RR estimator, in the same section. In Section 4, performances of where Z = XT, it implies that Z0Z ¼ K, and a ¼ T0b (see the proposed estimators with respect to the scalar MSE crite- Montgomery et al., 2001). rion compared to LS, RR and GR estimators are evaluated Then LS estimator of a is given by on basis of the Monte Carlo Simulation results. Influence of a^ Z0Z 1Z0Y K1Z0Y: 3 choice of k to compute RR on the proposed estimator AR is LS ¼ð Þ ¼ ð Þ also studied in the same section. Finally, article ends with some Therefore, LS estimator of b is given by concluding remarks. ^ bLS ¼ Ta^LS: 2. Model specifications and the estimators 2.1. Generalized ridge regression estimator (GR) We consider the linear regression model with p predictors and n observations: In order to combat multicollinearity and improve the LS esti- mator, Hoerl and Kennard (1970) suggested an alternative Y ¼ Xb þ e; ð1Þ estimator by adding a ridge parameter k to the diagonal ele-

0 0 ments of the least square estimator. They also suggested gener- where Y ¼ðY1; Y2; ...; YnÞ , b ¼ðb1; b2; ...; bpÞ , e ¼ðe1; e2; alized ridge regression (GR) estimator by using separate ridge 0 ...; enÞ and X ¼ðx1; x2; ...; xpÞ. ei’s are independently and parameter for each regressor in the ridge regression. Also, if identically distributed as normal with mean 0 and variance the optimal values for biasing constants differ significantly 2 r . Assume that the Yi’s are centered and the covariates xi’s from each other, then this estimator has the potential to save are standardized. Let K and T be the matrices of eigen values a greater amount of MSE than the LS estimator (Stephen and eigen vectors of X0X, respectively, satisfying and Christopher, 2001). 0 0 T X XT ¼ K ¼ diagonalðk1; k2; ...; kpÞ, where ki being the ith

Figure 2 ‘‘fm’’ for AR, RR and GR estimators (q = 0.99, p =3 Figure 4 ‘‘fm’’ for AR, RR and GR estimators (q = 0.9999, and b = (7, 4, 1, 8)0). p = 3 and b = (10, 1, 1, 4)0). 98 A.V. Dorugade

^ Hence GR estimator for b is bGR ¼ Ta^GR. and mean square error of a^GR is 2 MSEða^GRÞ¼Varianceða^GRÞþ½Biasða^GRÞ

Xp Xp 2 2 2 2 2 MSEða^GRÞ¼r^ ki=ðki þ kiÞ þ ki a^i =ðki þ kiÞ ð5Þ i¼1 i¼1

Setting k1 = k2 = ...= kp = k and k P 0, GR estimator reduces to RR estimator of a denoted by a^RR:. Hence, mean square error of a^RR is 1 a^RR ¼½I kðK þ kIÞ a^LS ð6Þ Therefore, RR estimator of b is given by ^ Figure 5 ‘‘fm’’ for AR, RR and GR estimators (q = 0.99999, bRR ¼ Ta^RR p = 3 and b = (8, 4, 11, 5)0). and mean square error of a^RR is The GR estimator of a is defined by Xp Xp 2 2 2 2 2 MSEða^RRÞ¼r^ ki=ðki þ kÞ þ k a^ =ðki þ kÞ ð7Þ 1 i a^GR ¼ðI KA Þa^LS; ð4Þ i¼1 i¼1 where K = diagonal(k1, k2...kp), ki P 0, i =1,2,...,p be the We observe that when k =0in(7), MSE of LS estimator of a different ridge parameters for different regressor and is recovered. Hence A = K + K.

0 Table 1 ‘‘fm’’ for AR, RR, GR and LS estimators (p = 4 and b = (2, 15, 3, 14, 8) ). q Estimator n = 20 50 100 500 r2 = 1 9 25 100 1 9 25 100 1 9 25 100 1 9 25 100 0.6 AR 1420 4810 5420 4930 180 3600 4080 4470 10 2010 4050 4070 0 170 1350 3010 RR 2420 2200 2970 4080 1950 2220 2630 3250 1850 2530 2140 2530 2020 2050 2180 2030 GR 4870 2530 1460 910 5870 3390 2730 2050 5650 4230 3110 2900 4930 5570 4920 3990 LS 1290 460 150 80 2000 790 560 230 2490 1230 700 500 3050 2210 1550 970 0.8 AR 2670 5980 6440 6020 700 4770 5570 5130 190 3700 4830 5330 0 700 2900 4180 RR 2400 2060 2610 3440 2280 2250 2260 3390 1470 2390 2340 2800 980 1610 2300 2430 GR 3950 1870 910 530 5370 2610 1920 1400 6730 3130 2450 1750 7190 6160 3930 2910 LS 980 90 40 10 1650 370 250 80 1610 780 380 120 1830 1530 870 480 0.9 AR 4190 6940 7460 7430 2080 5880 6580 6540 690 4970 6220 6430 0 2020 4390 5310 RR 2560 1750 1960 2200 2590 2010 2220 2770 1900 2070 1900 2250 900 2400 2340 1990 GR 2670 1220 540 350 4460 1950 1130 680 5990 2610 1720 1230 7390 4490 2790 2330 LS 580 90 40 20 870 160 70 10 1420 350 160 90 1710 1090 480 370 0.95 AR 5560 7810 8200 8080 3400 7080 7670 7580 1950 6190 7210 7500 0 3530 5410 6450 RR 2200 1480 1420 1660 2520 1650 1670 1960 2210 2000 1580 1720 1120 2630 1890 1570 GR 1940 660 370 260 3350 1170 650 460 4860 1590 1110 760 7640 3180 2270 1860 LS 300 50 10 0 730 100 10 0 980 220 100 20 1240 660 430 120 0.99 AR 7560 9140 9350 9680 6600 8670 9160 9270 5100 8120 8620 9170 1450 5970 7820 8330 RR 1580 620 580 280 1880 900 640 590 2620 1100 1020 680 2390 2220 1110 980 GR 810 240 60 40 1330 410 200 140 1920 730 360 150 5190 1590 950 670 LS 50 0 10 0 190 20 0 0 360 50 0 0 970 220 120 20 0.999 AR 9280 9780 9890 9900 8730 9640 9750 9850 8250 9410 9760 9850 6240 8690 9240 9530 RR 500 160 80 100 880 270 170 140 1150 410 180 70 2340 890 540 370 GR 220 60 30 0 360 90 80 10 560 180 60 80 1280 400 220 100 LS0 000 30000400001402000 0.9999 AR 9770 9910 9960 9980 9500 9820 9900 9960 9390 9810 9900 9930 8770 9530 9730 9830 RR 160 80 40 20 410 150 80 40 480 170 70 50 930 340 220 130 GR 70 10 0 0 90 30 20 0 130 20 30 20 260 130 50 40 LS0 000 0000000040000 0.99999 AR 9900 9990 9980 10,000 9830 9990 9940 9980 9780 9960 9930 9980 9590 9810 9930 9920 RR 80 10 20 0 120 10 60 20 190 30 60 20 310 140 40 60 GR20000 500003010100100503020 LS0 000 000000000000 Adjusted ridge estimator and comparison with Kibria’s method 99

0 Table 2 ‘‘fm1’’ and ‘‘fm2’’ for AR, RR, GR and LS estimators (p = 7 and b = (10, 1, 8, 5, 12, 1, 4, 7) ). q Estimator n = 20 100 500 r2 = 1 9 25 100 1 9 25 100 1 9 25 100

0.6 fm1 AR 5630 6140 5780 6010 3890 6210 5820 5880 50 4080 4960 3650 RR 2050 3050 3640 3390 2870 2230 3300 3750 2870 2250 2200 4020 GR 1800 750 560 600 2340 1410 840 370 4410 2500 2250 2240 LS 520 60 20 0 900 150 40 0 2670 1170 590 90

fm2 AR 1810 5600 6310 6070 80 3550 5470 6420 0 830 2710 3610 RR 4620 3210 2840 2340 5190 4410 3340 2980 3130 3970 3500 3530 GR 960 420 30 20 1220 720 620 140 2830 1510 1050 700 LS 2610 770 820 1570 3510 1320 570 460 4040 3690 2740 2160

0.99 fm1 AR 8840 9340 9550 9590 7190 8850 9040 9020 4570 8310 8750 8860 RR 800 560 340 330 1800 760 790 920 3020 910 800 890 GR 350 100 110 80 840 380 170 60 1940 720 410 250 LS 10 0 0 0 170 10 0 0 470 60 40 0

fm2 AR 2710 7450 8570 8490 310 4540 7020 7890 0 1340 4180 6020 RR 6330 2210 950 450 7720 4780 2680 1840 6200 7150 5150 3690 GR 170 70 20 10 270 210 150 50 710 140 220 90 LS 790 270 460 1050 1700 470 150 220 3090 1370 450 200

0.999 fm1 AR 9570 9910 9920 9900 9060 9690 9810 9930 8060 9180 9730 9730 RR 290 80 60 70 600 200 150 70 1370 570 220 210 GR 140 10 20 30 320 110 40 0 480 240 50 60 LS 0 0 0 0 20 0 0 0 90 10 0 0

fm2 AR 3180 7870 8310 8300 230 4590 6800 8430 0 1690 4250 6220 RR 5710 1970 970 450 8340 4870 3040 1450 8470 7260 5280 3520 GR 40 0 0 10 100 30 0 10 150 120 30 60 LS 1070 160 720 1240 1330 510 160 110 1380 930 440 200

0.9999 fm1 AR 9930 9980 10,000 9990 9750 9910 9950 9980 9280 9810 9950 9920 RR 50 20 0 0 190 60 30 20 550 130 50 80 GR 20 0 0 10 60 30 20 0 160 60 0 0 LS 0 0 0 0 0 0 0 0 10 0 0 0

fm2 AR 3070 7610 8420 8530 300 4510 6980 8160 0 1740 4220 6230 RR 5830 2260 890 370 8190 4930 2830 1600 8840 7060 5340 3540 GR0 2000 200100 1010010 LS 1100 110 690 1100 1490 560 180 240 1150 1190 440 220

! Xp Yp qffiffiffiffiffiffiffiffiffiffiffiffi ^ ^2 2 2 MSE ðaLSÞ¼r 1=ki ð8Þ k6 ¼ Median a^i =r^ ðMuniz and Kibria; 2009Þð14Þ i¼1 i¼1 There are different methods for estimating k that exists in Xp the present literature. However, following we listed some of 2 2 2 k7 ¼ pr^1 a^i where; r^1 the well known methods for choosing ridge parameter value i¼1 to compute RR estimator used in simulation study. Y0Y a^0 Z0Y ¼ LS ðKhalaf; 2012Þð15Þ n p 1 Xp 2 2 k1 ¼ pr^ a^i ðHoerl et al:; 1975Þð9Þ Also, in case of generalized ridge regression, the following i¼1 well known method for determination of ridge parameter for each regressor, given by Hoerl and Kennard (1970), is used r^2 k2 ¼ Q ðKibria; 2003Þð10Þ to compute a^GR . p a^2 1=p i¼1 i r^2 ki ¼ 2 ; i ¼ 1; 2; :::; p ð16Þ r^2 a^i k3 ¼ Median i ¼ 1; 2; :::; p ðKibria; 2003Þð11Þ 20 2 a^i where, a^i is the ith element of a^LS, i ¼ 1; 2; :::; p and r^ is the LS 0 ^0 0 2 2 Y YaLSZ Y 2 2 estimator of r i.e. r^ ¼ np1 : k4 ¼ðkmaxr^ Þ=ððn p 1Þr^ 2 þ kmaxa^maxÞðKhalaf and Shukur; 2005Þð12Þ 3. Proposed estimator ! qffiffiffiffiffiffiffiffiffiffiffiffi 1 Yp p The Ridge Regression (RR) estimator proposed by Hoerl and 2 2 k5 ¼ a^i =r^ ðMuniz and Kibria; 2009Þð13Þ Kennard (1970) is such an estimator widely used by statisti- i¼1 cians in the presence of multicollinearity. However, RR 100 A.V. Dorugade

0 Table 3 ‘‘fm’’ for AR and RR estimators for a different choice of k (p = 4 and b = (4, 2, 10, 1, 3) ). q Estimator kn= 20 100 500 r2 = 1 9 25 100 1 9 25 100 1 9 25 100 0.6 AR 40 2070 2810 1590 0 100 940 1950 0 0 10 410

RR k1 580 940 630 460 240 970 930 1080 60 330 660 860 k2 2570 2560 2280 1780 1640 2310 2710 2640 700 1660 2230 2780 k3 3390 2470 1650 1140 3770 3510 2930 2220 4460 3770 3370 3110 k4 230 580 430 430 100 350 440 550 70 120 270 340 k5 0 90 930 2610 0 0 10 130 0000 k6 0 60 350 1160 000900000 k7 0 00000000000 0.99 AR 2450 5300 4930 4290 240 3030 4570 5090 0 750 2100 3060

RR k1 700 210 110 90 960 620 300 210 520 1120 810 480 k2 2990 1680 1260 840 3430 2790 2510 1710 2690 3180 3230 2900 k3 2510 1500 1570 1570 3310 2200 1600 1420 4000 3070 2480 2170 k4 390 450 230 140 420 600 400 380 200 530 440 470 k5 0 350 1020 1780 0 0 110 500 0000 k6 0 190 700 1130 0003500000 k7 0 00000000000 0.999 AR 5660 7140 7150 7270 2830 6070 6920 7190 690 4020 5460 6590

RR k1 280 80 90 60 440 160 90 70 1060 500 240 100 k2 2000 750 660 490 3210 1360 1090 760 3270 2490 2020 1250 k3 1520 1770 1720 1490 2200 1620 1530 1760 3150 1950 1490 1570 k4 400 150 130 20 530 600 180 120 530 480 520 350 k5 0 013046000000000 k6 0 106014000000000 k7 0 00000000000 0.9999 AR 6790 7770 8410 8460 6110 7100 7540 7700 4440 6520 7050 7210

RR k1 100 10 20 30 130 130 60 50 470 70 110 100 k2 860 530 310 300 1890 780 630 500 2610 1240 750 840 k3 1970 1620 1190 1170 1420 1730 1620 1680 1570 1650 1810 1600 k4 120 20 30 20 300 130 50 60 470 360 190 150 k5 0 0102000000000 k6 0 00000000000 k7 0 00000000000

estimator has some disadvantages; mainly it is a nonlinear line of Y on X becomes the correlation coefficient. function of the ridge parameter (or biasing constant) k. Obliviously, Z0Y is the vector of correlation coefficient This leads to complicated equations, when k is selected. between Z and Y. By using the same vector with modification There is no explicit formula for this ridge parameter. Many in RR estimator, we proposed a new estimator of a which is authors proposed different approximations for it. The conven- termed as Adjusted Ridge (AR) Estimator and is given by: tional wisdom is that no single method would be uniformly hi 1 0 0 1=2 better than all the others. Also, as pointed out by Liu (2003) a^AR ¼½K þ C Z Y where; C ¼ diagonal ðÞjZ Yj when there exits sever multicollinearity the ridge parameter k or selected for ridge regression may not fully remedy the problem 1 of multicollinearity. To avoid calculating the value of k in this a^AR ¼½I CA a^LS where; A ¼ðK þ CÞ article, we suggest the modification in the Ridge Regression Hence, Adjusted Ridge Estimator of b is: (RR) estimator proposed by Hoerl and Kennard (1970) by ^ avoiding the determination of optimal ridge parameter k. bAR ¼ Ta^AR Now the idea is that the correlation coefficient between the regressors is helpful in detecting the near linear dependency between the same pairs of regressors only which plays an 3.1. Bias, variance and MSE of a^AR important role in detecting problem of multicollinearity. Rodgers and Nicewander (1988) present a longer review of Bias of a^AR: ways to interpret the correlation coefficient. Also, as inter- Biasða^ARÞ¼E½a^ARa preted by Nefzger and Drasgow (1957), for the bivariate data CA1 (X, Y) when we standardize the two raw variables, the stan- ¼ a dard deviations become unity and the slope of the regression Variance of a^AR: Adjusted ridge estimator and comparison with Kibria’s method 101 Âà 0 Varða^ARÞ¼E ða^AR Eða^ARÞÞ ða^AR Eða^ARÞÞ 0.90, 0.95, 0.99, 0.999, 0.9999 and 0.99999. Ten thousand sim- 2 1 0 ulations are run for all combinations of r = 1, 9, 25 100 and ¼ðI CA1Þr2K ðI CA1Þ n = 20, 50, 100 and 500. Here we used well known ridge 2 0 1 2 1 where MSEða^LSÞ¼Vða^LSÞ¼r ðZ ZÞ ¼ r K parameter k1 given by Hoerl et al. (1975). MSE of estimators MSE of a^AR: computed using the following expression, Xp 2 ^ ^ 2 MSEða^ARÞ¼Vða^ARÞþ½Biasða^ARÞ MSEðbÞ¼ ðbi biÞ 0 ¼ðI CA1Þr2K1ðI CA1Þ þ CA1aa0CA1 i¼1 ^ .where, bi denote the estimator of the ith parameter and bi, Xp 2 2 i ¼ 1; 2; ...; p are the true parameter values. However, b r ki þðaiciÞ MSEða^ARÞ¼ 2 parameter vectors are chosen arbitrarily for number of regres- i¼1 ðki þ ciÞ sors p=4. For each simulation, the dependent variables are computed by the specified protocol. Values of ‘‘fm’’ reported where, ci is the ith diagonal element of C, i ¼ 1; 2; ...; p in Table 1, indicate the frequency with which each estimator Or had the lowest MSEðb^Þ. We consider the method that leads Xp 2 to the maximum ‘‘fm’’ to the best from the MSE point of view. r^2k þða^ c Þ i i i The same procedure above for another choice of p = 3 and MSEða^ARÞ¼ 2 ð17Þ i¼1 ðki þ ciÞ arbitrarily chosen parameter vectors b are done and values of

2 ‘‘fm’’ are computed and represented in Figures. Here we noted where, a^i is the ith element of a^LS, i ¼ 1; 2; ...; p and r^ are the 0 ^0 0 that values of ‘‘fm’’ only for RR, GR and AR are represented 2 2 Y YaLSZ Y LS estimator of r i.e. r^ ¼ np1 : because these values for LS have less importance for the com- parative study. To compute and represent the results with

3.2. Comparison between the a^AR and a^RR respect to the presence of moderate or extremely sever multi- collinearity, we consider q = 0.95, 0.99, 0.999, 0.9999 and 2 It is well known that, the value of ridge parameter ‘k’ is chosen 0.99999. Here input values are n and r . These input values small enough, for which the mean squared error of RR estima- are ordered according to the increase of values. For fixed value 2 tor, is less than the mean squared error of LS estimator. Also of ‘n’ changes the values of r . 2 most of the researchers studied comparison between RR and There are 16 sets of (n, r ) values. These are arranged as GR estimators. Hence, in the following, we compare our pro- (20, 1), (20, 9),..., (500, 100) and it is numbered as 1, 2,..., posed estimator to the RR estimator only. Using (7) and (17) 16 respectively. Obtained results are represented in Figs. 1–5. we investigate the following difference In addition to demonstrate the other performances of the  proposed method, we have computed the relative error sum Xp Xp r^2k þk2 a^2 2 2 ðÞi i r^ ki þða^i ci Þ of squares of parameters (RESSðbÞ) as well as prediction mean MSEða^RRÞMSEða^ARÞ¼ 2 2 ðki þkÞ ðkiþci Þ i¼1 i¼1 square error (MSEðyÞ) to show the predicting ability of the p X 2 2 2 2 2 2 2 ½ðkiþci Þ ðr^ ki þk a^i Þðr^ ki þða^i ci Þ ÞðkiþkÞ model developed by the proposed and the other estimators ¼ 2 2 ðkiþci Þ ðkiþkÞ using following expressions. i¼1 p X 2 2 2 2 2 2 2 2 fg½ðki þci Þ ðkiþkÞ r^ kiþð½ki þciÞ k ðciÞ ðki þkÞ a^i Xp ¼ 2 2 2 ðki þciÞ ðki þkÞ i¼1 MSEðyÞ¼ ðy y^Þ i¼1 2 2 Since the quantity ðki þ ciÞ ðki þ kÞ is always positive, "# Xp 2 from above equation, it can be shown that MSEða^ Þ P ðb b^Þ RR RESSðbÞ¼ 2 2 2 2 b MSEða^ARÞ if and only if ðki þ ciÞ k P ðciÞ ðki þ kÞ . i¼1 Results on ‘‘f ’’ and ‘‘f ’’ which are reported in Table 2, 4. Simulation study m1 m2 indicate the frequencies with which each estimator had the lowest RESSðbÞ and MSEðyÞ; respectively. We consider the We are now ready to illustrate the behavior of the proposed method that leads to the maximum ‘‘fm1’’ and ‘‘fm2’’ to the best estimator via a Monte Carlo simulation. We performed our from the MSE point of view. simulations with MATLAB, using different sample sizes and Tables 1 and 2 indicate that when multicollinearity is error variances examined the MSE of the estimators LS, RR, nonexistent with lower error variance r2 only (at q = 0.6 GR and AR for different degrees of multicollinearity. For and r2 = 1) the improvement of AR is not very substantial, the simulations, we supposed the regression model defined in since in this case RR and GR are themselves fine estimators. Eq. (1). Following McDonald and Galerneau (1975) the However, when multicollinearity is moderate or sever or extre- explanatory variables are generated by mely sever, the improvement is extremely effective and dra- 1=2 matic, because in this case not only LS but also RR and GR x ¼ð1 q2Þ u þ qu ; i ¼ 1; 2; :::; nj¼ 1; 2; :::; p: ij ij ip perform very poorly as shown by the simulation. Especially where, uij are independent standard normal pseudo-random under the situation, when multicollinearity is extremely sev- numbers and q is specified so that the theoretical correlation ered; the level of multicollinearity influences the improvement between any two explanatory variables is given by q2. In this of the AR over other estimators. Similarly, increasing the error study, to investigate the effects of different degrees of multi- variance seems to improve the accuracy of AR. However, in collinearity on the estimators, we consider, q = 0.6, 0.8, Tables 1 and 2, it is also seen that increasing the sample size 102 A.V. Dorugade at lower error variance for nonexistent or moderate multi- References collinearity GR is superior to LS, RR and AR estimators. But, for sever or extremely sever multicollinearity AR is supe- Batah, F.S., Ramnathan, T., Gore, S.D., 2008. The efficiency of rior to others for large sample size, even error variances are modified jackknife and ridge type regression estimators: a compar- small. For sever or extremely sever multicollinearity AR is ison. Surv. Math. App. 24 (2), 157–174. consistently superior to LS, RR and GR estimators for differ- Dorugade, A.V., 2014. New ridge parameters for ridge regression. J. ent combinations of size of the sample (n), variance of the error Assoc. Arab Univ. Basic Appl. Sci. 15, 94–99. term (r2) and number of predictors (p). Two novel features of Hoerl, A.E., Kennard, R.W., 1970. 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John Wiley and Sons, New York. p=4 and values of ‘‘fm’’ are computed and reported in Muniz, G., Kibria, B.M.G., 2009. On some ridge regression estima- Table 3. We consider the method that leads to the maximum tors: an empirical comparison. Commun. Stat. Simul. Comput. 38, 621–630. ‘‘fm’’ to the best from the MSE point of view. Table 3, clearly indicates that choice of k to compute RR does not influence Nefzger, M.D., Drasgow, J., 1957. The needless assumption of the performance of the proposed estimator AR. normality in Pearson’s r. Am. Psychol. 12, 623–625. Rodgers, J.L., Nicewander, W.A., 1988. Thirteen ways to look at the correlation coefficient. Am. Stat. 42 (1), 59–66. 5. Conclusion Saleh, A.K.Md.E., Kibria, B.M.G., 1993. Performances of some new preliminary test ridge regression estimators and their properties. Commun. Stat. 22, 2747–2764. This article introduces a new method for regression parameter Sarkar, N., 1992. A new estimator combining the ridge regression and estimation which aims at totally avoiding computational part the restricted least squares method of estimation. Commun. Stat. for optimal ridge parameter k in ridge regression. Our 21, 1987–2000. suggested estimator is termed as AR since it is obtained by Singh, B., Chaubey, Y.P., 1987. On some improved ridge estimators. adjusting RR estimator, given by Hoerl and Kennard (1970). Stat. Pap. 28, 53–67. New estimator is a better alternative to RR and GR estimators Stephen, G.W., Christopher, J.P., 2001. Generalized ridge regression in the presence of sever or extremely sever multicollinearity and a generalization of the Cp statistic. J. Appl. Stat. 28 (7), 911– 922. with increasing error variance in linear regression. We believe Swindel, B.F., 1976. Good ridge estimators based on prior informa- that AR is a fine estimator, not only in theory but also in tion. Commun. Stat. Theor. Methods 11, 1065–1075. practice. Yan, X., 2008. Modified nonlinear generalized ridge regression and its application to develop naphtha cut point soft sensor. Comput. Acknowledgements Chem. Eng. 32 (3), 608–621. Yan, X., Zhao, W., 2009. Concentration soft sensor based on modified bak propagation algorithm embedded with ridge regression. Intell. The author wishes to thank the anonymous referee for his/her Autom. Soft Comput. 15 (1), 41–51. comments and the editor who improved the original version. Yang, H., Chang, X., 2010. A new two-parameter estimator in linear The present studies were supported in part Research Award regression. Comm. Stat. Theor. Methods 39, 923–934. Scheme by UGC, India Project No. F. 30-1/2014/RA-2014- Zhong, Z., Yang, H., 2007. Ridge estimation to the restricted linear 16-GE-MAH-5958 (SA-II). model. Commun. Stat. Theor. Methods 36, 2099–2115. Printed at UoB Printing Press‐2017‐104282 Journal of the Association of Arab Universities for Basic and Applied Sciences (JAAUBAS)

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