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Global Institute Lahore Global Institute Lahore HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION BY GHULAM FARID DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS February, 2015 GLOBAL INSTITUTE LAHORE i HADAMARD k-FRACTINAL INTEGRAL AND ITS APPLICATION BY GHULAM FARID A dissertation submitted to Faculty of Management Sciences In Partial Fullfilment of the Requirements for the Degree of DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS FEBRUARY, 2015 ii IN THE NAME OF ALLAH, THE MOST BENIFICIAL, THE MOST MERCIFUL iii GLOBAL INSTITUTE LAHORE HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION BY GHULAM FARID A dissertation submitted to Faculty of Management Sciences, in partial fullfilment of the requirements for the degree of DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS Dissertation Committee iv DECLARATION This is to certify that this research work has not been submitted for obtaining similar degree from any other university/college. v DEDICATED TO My Beloved Son Muhammad Yahya Khan vi ACKNOWLEDGEMENT Gratitude to almighty Allah for His unending blessings that has reinforced me to complete this humble piece of dissertation. I am also obliged to all the teachers and fellows at Global Institute, Lahore. I am grateful to my supervisor Dr. Ghulam Mustafa Habibullah for his motivation, support and guidance to the right direction. I am highly obliged to Muhammad Kashif Khan, the Chairman Global Institute Lahore and all of the college officials including the Rector, Pro-Rector and all the faculty members of the institute for their guidance, support and favours. The financial support of Higher Education Commission, Pakistan in facilitating me to take up this research work, is most gratefully acknowledged. I am also thankful to my brother Professor Muhammad Zulqarnain Khan and my wife for their continuous support and encouragement. At the end, I pay tribute to my parents and elders who always supported me and prayed for my success. vii RESEARCH COMPLETION CERTIFICATE Certified that the research work contained in this thesis titled "HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION" has been carried out and completed by Ghulam Farid under my supervision during his PhD Business Mathematics Programme. (Prof. Dr. G. M. Habibullah) Supervisor viii SUMMARY The Fractional Calculus has been attractive and hot topic among the researchers since 18th century, because of its extensive application in differential and integral equations and other disciplines of mathematics, physics and economics. The motivation of this thesis is to extend the fractional integrals and derivatives, particularly Hadamard fractional integral, and to establish basic properties of the extended fractional integral operators. The application of the extended operators involving the formation of the fractional integral inequalities and solutions of fractional integral equations is focussed in the work. The first chapter includes the introductory background of the fractional calculus. The appropriate literature pertaining to the fractional calculus, involving the theoretical and practical aspects of fractional differential and fractional integral operators has been reviewed. In the second chapter, we have listed symbols, notations and the basic results that are used throughout the dissertation. A number of inequalities involving the Holder’s inequality and AM-GM inequality have been presented. We have defined an extended form of Hadamard fractional integral and have called it Hadamard k-fractional integral. We have also discussed a number of properties of the extended integral operator. In the third chapter, we have established numerous fractional integral inequalities involving the inequalities of Chebyshev functional using the notion of synchronous functions, asynchronous functions, and the like. In the fourth chapter, we have presented some inequalities involving the rearrangement inequalities. On the basis of AM-GM, Holder and the rearrangement inequalities, we have established many fractional integral inequalities related to the extended operator. In the fifth chapter, we have introduced a number of extensions of the fractional integral operators involving the Hadamard type fractional operators. We have discussed the properties of the extended operators involving the semigroup property and commutative law. We have also considered the Mellin transforms and boundedness of some of the extended operators. In the sixth chapter, we have introduced extended fractional derivatives related to the extended fractional integral operators and have discussed their compositions. In the seventh chapter, we have presented some integral equations and have found their solutions using some of the extended fractional integral operators. We have also illustrated the use of some of the extended fractional calculus operators in finding solutions of fractional differential equations. ix TABLE OF CONTENTS Page DECLARATION v DEDICATION vi ACKNOWLEDGEMENT vii RESEARCH COMPLETION CERTIFICATE viii SUMMARY ix CHAPTER 1 1 INTRODUCTION 1.1 BACKGROUND 1 1.2 LITERATURE REVIEW 1 CHAPTER 2 11 EXTENSION OF HADAMARD FRACTIONAL INTEGRAL 2.1 PRELIMINARIES 11 2.2 HADAMARD k-FRACTIONAL INTEGRAL 19 2.2.1 Definition 19 2.3 PROPERITES OF HADAMARD k-FRACTIONAL INTEGRAL 19 2.3.1 Theorem 19 2.3.2 Corollary 20 2.3.3 Theorem (Commutative Law) 20 2.3.4 Theorem (Semigroup Property) 21 2.3.5 Definition 21 2.3.6 Theorem 21 2.3.7 Theorem 22 2.3.8 Theorem 22 2.3.9 Theorem 22 2.3.10 Theorem 23 2.3.11 Theorem 23 2.3.12 Theorem 24 k 2.3.13 Theorem (Boundedness of HI(,) a t ) 24 CHAPTER 3 25 FI INEQUALITIES RELATING THE EXTENDED OPERATOR 3.1 FI INEQUALITIES FOR CHEBYSHEV FUNCTIONAL 25 3.1.1 Theorem 26 3.1.2 Theorem 26 3.1.3 Note 26 3.1.4 Note 27 3.1.5 Note 27 3.1.6 Note 27 3.1.7 Theorem 27 x 3.1.8 Lemma 27 3.1.9 Corollary 28 3.1.10 Note 28 3.1.11 Lemma 28 3.1.12 Lemma 29 3.1.13 Note 29 3.1.14 Theorem 29 3.1.15 Theorem 30 3.1.16 Corollary 31 3.1.17 Theorem 31 3.1.18 Theorem 32 3.1.19 Corollary 32 3.1.20 Theorem 32 3.1.21 Corollary 33 3.1.22 Theorem 33 3.1.23 Theorem 34 3.1.24 Theorem 35 3.1.25 Theorem 35 3.1.26 Theorem 36 3.1.27 Theorem 37 3.1.28 Theorem 37 3.1.29 Theorem 38 3.1.30 Theorem 39 3.1.31 Note 39 3.1.32 Theorem 39 3.1.33 Theorem 40 3.1.34 Theorem 41 3.1.35 Theorem 42 CHAPTER 4 43 MORE FI INEQUALITIES FOR THE EXTENDED OPERATOR 4.1 GRUSS TYPE INEQUALITIES AND FI INEQUALITIES RELATING AM-GM AND HOLDER’S INEQUALITIES 43 4.1.1 Theorem 43 4.1.2 Corollary 44 4.1.3 Theorem 45 4.1.4 Theorem 45 4.1.5 Theorem 46 4.1.6 Corollary 47 4.1.7 Theorem 47 4.1.8 Example 47 4.1.9 Corollary 48 4.1.10 Theorem 48 4.1.11 Corollary 49 xi 4.1.12 Theorem 49 4.1.13 Example 50 4.1.14 Corollary 51 4.1.15 Theorem 51 4.1.16 Example 52 4.1.17 Corollary 53 4.1.18 Theorem 53 4.1.19 Corollary 54 4.1.20 Theorem 54 4.1.21 Corollary 55 4.1.22 Theorem 55 4.1.23 Corollary 56 4.1.24 Theorem 56 4.1.25 Corollary 57 4.1.26 Theorem 57 4.1.27 Theorem 57 4.1.28 Theorem 58 4.1.29 Corollary 59 4.2 FI INEQUALITIES BASED ON REARRANGEMENT INEQUALITIES 59 4.2.1 Lemma 59 4.2.2 Note 60 4.2.3 Example 60 4.2.4 Note 60 4.2.5 Theorem 60 4.2.6 Theorem 61 4.2.7 Corollary 62 4.2.8 Example 62 4.2.9 Theorem 62 4.2.10 Theorem 63 4.2.11 Lemma 65 4.2.12 Note 65 4.2.13 Lemma 65 4.2.14 Note 66 4.2.15 Example 66 4.2.16 Theorem 66 4.2.17 Theorem 67 4.2.18 Theorem 68 4.2.19 Example 69 4.2.20 Theorem 70 4.2.21 Lemma 70 4.2.22 Theorem 72 4.2.23 Example 73 xii 4.2.24 Example 73 4.3 MORE FI INEQUALITIES RELATING AM-GM AND HOLDER’S INEQUALITIES 74 4.3.1 Theorem 74 4.3.2 Example 75 4.3.3 Corollary 75 4.3.4 Corollary 76 4.3.5 Lemma 76 4.3.6 Corollary 77 4.3.7 Theorem 78 4.3.8 Theorem 78 4.3.9 Example 79 4.3.10 Theorem 81 4.3.11 Theorem 82 4.3.12 Corollary 83 4.3.13 Theorem 84 4.3.14 Theorem 87 4.3.15 Theorem 90 4.3.16 Theorem 91 CHAPTER 5 93 MORE EXTENSIONS OF FI OPERATORS 5.1 EXTENSIONS OF THE FRACTIONAL INTEGRAL OPERATOR 93 5.2 PROPERTIES OF THE EXTENDED OPERATORS 94 5.2.1 Theorem 94 5.2.2 Corollary 95 5.2.3 Theorem 95 5.2.4 Corollary 96 5.2.5 Theorem 96 5.2.6 Corollary 96 5.2.7 Theorem 97 5.2.8 Corollary 97 5.2.9 Lemma 97 5.2.10 Lemma 98 5.2.11 Lemma 99 5.2.12 Theorem 100 5.2.13 Theorem 100 5.2.14 Theorem 101 5.2.15 Theorem 101 5.2.16 Lemma 102 5.2.17 Theorem 102 5.2.18 Corollary 103 5.2.19 Theorem 103 xiii 5.2.20 Corollary 104 5.3 OTHER EXTENDED FRACTIONAL INTEGRALS 104 5.3.1 Theorem 104 5.3.2 Theorem 104 5.3.3 Theorem 105 5.3.4 Corollary 105 5.3.5 Theorem 105 5.3.6 Corollary 106 5.4 MELLIN TRANSFORM OF THE EXTENDED FRACTIONAL INTEGRALS 106 5.4.1 Lemma 106 5.4.2 Theorem 106 5.4.3 Theorem 107 5.4.4 Theorem 108 5.4.5 Theorem 109 5.5 BOUNDEDNESS OF THE EXTENDED FRACTIONAL INTEGRAL OPERATORS 110 5.5.1 Definition 110 5.5.2 Theorem 111 5.5.3 Corollary 113 5.5.4 Corollary 113 CHAPTER 6 115 THE EXTENDED FD OPERATORS 6.1 FRACTIONAL DERIVATIVE RELATED TO HADAMARD k- FRACTIONAL INTEGRAL 115 6.1.1 Definition 115 6.1.2 Theorem 115 6.1.3 Corollary 116 6.2 FRACTIONAL DERIVATIVES RELATED TO OTHER EXTENDED FRACTIONAL INTEGRALS 116 6.2.1 Definition 116 6.2.2 Theorem 117 6.2.3 Corollary 117 6.2.4 Definition 118 6.2.5 Theorem 118 6.2.6 Corollary 119 6.2.7 Definition 119 6.2.8 Theorem 119 6.2.9 Corollary 120 6.2.10 Definition 120 6.2.11 Theorem 120 6.2.12 Corollary 121 xiv CHAPTER 7 122 APPLICATION OF THE EXTENDED FC OPERATORS 7.1 SOLUTIONS OF INTEGRAL EQUATIONS USING HADAMARD k-FRACTIONAL INTEGRAL OPERATOR 122 7.1.1 Lemma 122 7.1.2 Corollary 123 7.1.3 Lemma 123 7.1.4 Theorem 124 7.1.5 Theorem 124 7.2 SOLUTIONS OF INTEGRAL EQUATIONS USING ANOTHER EXTENDED FI OPERATOR 125 7.2.1 Lemma 125 7.2.2 Lemma 125 7.2.3 Theorem 126 7.2.4 Theorem 126 7.3 SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS VIA EXTENDED FC OPERATORS 127 7.3.1 Example 127 CONCLUSION 129 RECENTLY PUBLISHED PAPERS OF THE AUTHOR 130 REFERENCES 131-138 xv CHAPTER 1 INTRODUCTION 1.1 BACKGROUND Calculus has a long historical background, however the revolution of modern calculus came in 17th century.
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