DOKUZ EYLÜL ÜNİVERSİTESİ MÜHENDİSLİK FAKÜLTESİ

FEN VE MÜHENDİSLİK DERGİSİ Cilt/Vol.:18■No/Number:3■Sayı/Issue:54■Sayfa/Page:279-289■EYLÜL 2016/Sep 2016 DOI Numarası (DOI Number): 10.21205/deufmd.2016185401

Makale Gönderim Tarihi (Paper Received Date): 20.01.2016 Makale Kabul Tarihi (Paper Accepted Date): 02.04.2016

EXPLICIT SOLUTIONS OF THE CONFLUENT HYPERGEOMETRIC EQUATIN BY MEANS OF THE DIFFERINTEGRAL THEOREMS

(DİFERİNTEGRAL TEOREMLERİ YARDIMIYLA KONFLUENT HİPERGEOMETRİK DENKLEMİNİN AÇIK ÇÖZÜMLERİ)

Ökkeş ÖZTÜRK1

ABSTRACT In fractional , an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Differintegral theory is used to solve some classes of differential equations and fractional differential equations. One of these equations is the confluent hypergeometric equation. In this paper, we intend to solve this equation by means of the differintegral theorems.

Keywords: , Differintegral, Confluent hypergeometric equation, Differintegral theorems, Generalized Leibniz rule

ÖZ Uygulamalı matematiğin bir alanı olan kesirli hesapta diferintegral, türev/ operatörünün bir birleşimidir. Diferansiyel denklemlerin ve kesirli diferansiyel denklemlerin bazı sınıflarını çözmek için diferintegral teorisi kullanılmaktadır. Bu denklemlerden birisi konfluent hipergeometrik denklemidir. Bu makalede, diferintegral teoremleri yardımıyla bu denklemi çözmeyi hedefleriz.

Anahtar Kelimeler: Kesirli hesap, Diferintegral, Konfluent hipergeometrik denklemi, Diferintegral teoremleri, Genelleştirilmiş Leibniz kuralı

1 Bitlis Eren Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, BİTLİS, [email protected] (Sorumlu Yazar) Sayfa No: 280 Ö. ÖZTÜRK

1. INTRODUCTION

The widely investigated subject of fractional calculus (that is, calculus of and of any arbitrary real or complex order) has gained considerable importance and popularity during the past three decades or so, due chiefly to its demonstrated applications in numerous seemingly diverse fields of science and engineering. We can mention that the fractional differential equations are playing an important role in fluid dynamics, traffic model with fractional , measurement of viscoelastic material properties, modeling of viscoplasticity, control theory, relativity theory, economy, nuclear magnetic resonance, geometric mechanics, mechanics, optics, signal processing, robot technology, PID control systems, Schrödinger equation, heat transfer, filtration and so on.

Some of most obvious formulations based on the fundamental definitions of Riemann- Liouville fractional differentiation and fractional integration are, respectively,

푡 1 푑푘 퐷휇푓(푡) = ∫ 푓(휏)(푡 − 휏)푘−휇−1 푑휏 (푘 − 1 ≤ 휇 < 푘), (1) 푎 푡 Г(푘 − 휇) 푑푡푘 푎 and,

푡 1 퐷−휇푓(푡) = ∫ 푓(휏)(푡 − 휏)휇−1 푑휏 (푡 > 푎, 휇 > 0), (2) 푎 푡 Г(휇) 푎 where 푘 ∈ ℕ, ℕ being the set of positive integers, Γ stands for Euler’s gamma [1-4].

Recently, by applying the Riemann-Liouville definitions of a differintegral (that is, fractional derivative and fractional integral) of order 휇 ∈ ℝ, many authors have explicity obtained particular solutions of a number of families of homogeneous (as well as non- homogeneous) linear ordinary and partial differintegral equations (see, for details, [5]; see also [6,7]). An important example of Fuchsian differential equations is provided by the celebrated hypergeometric equation (or, more precisely, the Gauss hypergeometric equation)

푑2푢 푑푢 푧(1 − 푧) + [훾 − (훼 + 훽 + 1)푧] − 훼훽푢 = 0, 푑푧2 푑푧 whose study can be traced back to L. Euler, C.F. Gauss and E.E. Kummer. On the other hand, a special limit (confluent) case of the Gauss hypergeometric equation, in the form [8]

푑2푢 1 휘 ℓ(ℓ + 1) 1 + (− + − ) 푢 = 0 (휇 = ℓ + ), 푑푧2 4 푧 푧2 2 is refered to as the Whittaker equation whose systematic study was initiated by E.T. Whittaker.

Other classes of non-Fuchsian differential equations which we shall consider in this investigation include the so-called Fukuhara equation [9]

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푑2푢 푑푢 푧2 + 푧 − (1 − 푧 + 푧2)푢 = 0, 푑푧2 푑푧 the Tricomi equation [10]

푑2푢 훽 푑푢 훿 휀 + (훼 + ) + (훾 + + ) 푢 = 0, 푑푧2 푧 푑푧 푧 푧2 and the Bessel equation [11]

푑2푢 푑푢 푧2 + 푧 − (푧2 − 푣2)푢 = 0. 푑푧2 푑푧

Moreover, in [12], Inc obtained the particular solutions of the confluent hypergeometric differential equation by using the nabla fractional calculus operator which is an important operator in discrete fractional calculus. Virchenko’s study [13] is devoted to further development of important case of Wright’s hypergeometric function and its applications to the generalization of Γ −, 퐵 −, 휓 −, 휁 −, Volterra functions. In [14], Srivastava and Saxena expressed some Volterra-type fractional integro-differential equations with a multivariable confluent hypergeometric function as their kernel. And, Campos solved the extended confluent hypergeometric differential equation in [15].

In this paper, we also obtained the fractional solutions of the confluent hypergeometric equation by using the differintegral theorems. The most important advantage of these theorems is applicaple to the singular equations.

2. MATERIALS AND METHODS

2.1. Definition If the function 푓(푧) is analytic (regular) inside and on 퐶, where 퐶 = {퐶−, 퐶+}, 퐶− is a contour along the cut joining the points 푧 and −∞ + 𝑖Im(푧), which starts from the point at −∞, encircles the point 푧 once counter-clockwise, and returns to the point at −∞, and 퐶+ is a contour along the cut joining the points 푧 and ∞ + 𝑖Im(푧), which starts from the point at ∞, encircles the point 푧 once counter-clockwise, and returns to the point at ∞,

Г(휇 + 1) 푓(휏)푑휏 푓 (푧) = [푓(푧)] = ∫ (휇 ∉ ℤ−), 휇 휇 2휋𝑖 (휏 − 푧)휇+1 (3) 퐶 + 푓−푘(푧) = lim 푓휇(푧) (푘 ∈ ℤ ), 휇→−푘 where 휏 ≠ 푧,

−휋 ≤ arg(휏 − 푧) ≤ 휋 for 퐶−, (4) 0 ≤ arg(휏 − 푧) ≤ 2휋 for 퐶+.

In that case, 푓휇(푧) (휇 > 0) is the fractional derivative of 푓(푧) of order 휇 and 푓휇(푧) (휇 < 0) is the fractional integral of 푓(푧) of order −휇, confirmed (in each case) that

|푓휇(푧)| < ∞ (휇 ∈ ℝ). (5)

Sayfa No: 282 Ö. ÖZTÜRK

[4].

2.2. Lemma (Linearity) Let 푓(푧) and 푔(푧) be analytic and single-valued functions. If 푓휇 and 푔휇 exist, then

(퐢) [푐 푓(푧)] = 푐 [푓(푧)] , 1 휇 1 휇 (6) (퐢퐢) [푐1푓(푧) + 푐2푔(푧)]휇 = 푐1[푓(푧)]휇 + 푐2[푔(푧)]휇, where 푐1 and 푐2 are constants and 휇 ∈ ℝ, 푧 ∈ ℂ.

2.3. Lemma (Index law) Let 푓(푧) be an analytic and single-valued function. If (푓휈)휇 and (푓 ) exist, then 휇 휈

{[푓(푧)] } = [푓(푧)] = {[푓(푧)] } , (7) 휈 휇 휈+휇 휇 휈

Г(휈+휇+1) where 휈, 휇 ∈ ℝ, 푧 ∈ ℂ and | | < ∞. Г(휈+1)Г(휇+1)

2.4. Lemma (Generalized Leibniz rule) Let 푓(푧) and 푔(푧) be single-valued and analytic functions. If 푓휇 and 푔휇 exist, then

∞ Г(휇 + 1) (푓. 푔) = ∑ 푓 . 푔 , (8) 휇 Г(휇 + 1 − 푘)Г(푘 + 1) 휇−푘 푘 푘=0

Г(휇+1) where 휇 ∈ ℝ, 푧 ∈ ℂ and | | < ∞. Г(휇+1−푘)Г(푘+1)

2.5. Property For a constant 휆,

(e휆z) = 휆휈e휆푧 (휆 ≠ 0, 휈 ∈ ℝ, z ∈ ℂ). 휈 (9)

2.6. Property For a constant 휆,

(e−휆푧) = e−푖휋휈휆휈e−휆푧 (휆 ≠ 0, 휈 ∈ ℝ, 푧 ∈ ℂ). (10) 휈

2.7. Property For a constant 휆,

Γ(휈 − 휆) Γ(휈 − 휆) (푧휆) = e−푖휋휈푧휆−휈 (휈 ∈ ℝ, 푧 ∈ ℂ, | | < ∞). (11) 휈 Γ(−휆) Γ(−휆)

2.8. Property

Г(푧 + 1) = 푧Г(푧 + 1) = 푧!, (12) and,

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Γ(휈)Γ(1 − 휈) Γ(휈 − 푘) = (−1)푘 , (13) Γ(푘 + 1 − 휈)

+ where 푘 ∈ ℤ0 and 휈 ∈ ℝ.

2.9. Theorem Let 풫(푧; 퓅) and 풬(푧; 퓆) be polynomials in 푧 of degrees 퓅 and 퓆, respectively, defined by

퓅 퓅 퓅−푘 풫(푧; 퓅) = ∑ 푎푘푧 = 푎0 ∏(푧 − 푧푗) (푎0 ≠ 0, 퓅 ∈ ℕ), (14) 푘=0 푗=1 and,

퓆 퓆−푘 풬(푧; 퓆) = ∑ 푏푘푧 (푏0 ≠ 0, 퓆 ∈ ℕ). (15) 푘=0

Suppose also that 푓−휇 ≠ 0 exists for a given function 푓.

Then the nonhomogeneous linear ordinary fractional differintegral equation

퓅 퓆 휇 휇 풫(푧; 퓅)휑 (푧) + [∑ ( ) 풫 (푧; 퓅) + ∑ ( ) 풬 (푧; 퓆)] 휑 (푧) 휈 푘 푘 푘 − 1 푘−1 휈−푘 푘=1 푘=1

휇 + ( ) 퓆! 푏 휑 (푧) = 푓(푧) (퓅, 퓆 ∈ ℕ, 휈, 휇 ∈ ℝ), (16) 퓆 0 휈−퓆−1 has a particular solution of the form

푓−휇(푧) 휑(푧) = {[ eℋ(푧;퓅,퓆)] e−ℋ(푧;퓅,퓆)} (푧 ∈ ℂ ⧵ {푧 , … , 푧 }), 풫(푧; 퓅) 1 퓅 (17) −1 휇−휈+1 where for suitable condition,

푧 풬(휉; 퓆) ℋ(푧; 퓅, 퓆) = ∫ 푑휉 (푧 ∈ ℂ ⧵ {푧 , … , 푧 }), (18) 풫(휉; 퓅) 1 퓅 confirmed that the second component of (17) exists. Moreover, the homogeneous linear ordinary fractional differintegral equation

퓅 퓆 휇 휇 풫(푧; 퓅)휑 (푧) + [∑ ( ) 풫 (푧; 퓅) + ∑ ( ) 풬 (푧; 퓆)] 휑 (푧) 휈 푘 푘 푘 − 1 푘−1 휈−푘 푘=1 푘=1

Sayfa No: 284 Ö. ÖZTÜRK

휇 + ( ) 푞! 푏 휑 (푧) = 0 (퓅, 퓆 ∈ ℕ, 휈, 휇 ∈ ℝ), (19) 퓆 0 휈−퓆−1 has solutions of the form

휑(푧) = 퐾[e−ℋ(푧;퓅,퓆)] , 휇−휈+1 (20) where ℋ(푧; 퓅, 퓆) is given by (18), it being confirmed that the second component of (20) exist and 퐾 is an arbitrary constant [16].

3. MAIN RESULTS

The hypergeometric equation

푑2휑(푥) 푑휑(푥) 푥(1 − 푥) + [푐 − (푎 + 푏 + 1)푥] − 푎푏휑(푥) = 0, (21) 푑푥2 푑푥 has three regular singular points at 푥 = 0,1 and ∞ (푎, 푏 and 푐 are parameters). By setting 푥 = 푧⁄푏 and taking the limit as 푏 → ∞, we can merge the singularities at 푏 and infinity. This gives us the confluent equation as

푑2휑 푑휑 푧 + (푐 − 푧) − 푎휑 = 0, (22) 푑푧2 푑푧 solutions of which are the confluent hypergeometric functions, which are shown as 푀(푎, 푐; 푧).

The confluent hypergeometric equation has a regular singular point at 푧 = 0 and an essential singularity at infinity. Bessel functions, 퐽푛(푧), and the Laguerre polynomials, 퐿푛(푧), can be written in terms of the solutions of the confluent hypergeometric equation as

e−푖푧 푧 푛 1 퐽 (푧) = ( ) 푀 (푛 + , 2푛 + 1; 2𝑖푧), 푛 푛! 2 2 퐿푛(푧) = 푀(−푛, 1; 푧).

Linearly independent solutions of Eq. (22) are given as

푎 푧 푎(푎 + 1) 푧2 푎(푎 + 1)(푎 + 2) 푧3 휑 (푧) = 푀(푎, 푐; 푧) = 1 + + + + ⋯, 1 푐 1! 푐(푐 + 1) 2! 푐(푐 + 1)(푐 + 2) 3! (푐 ≠ 0, −1, −2, … ), and,

1−푐 휑2(푧) = 푧 푀(푎 + 1 − 푐, 2 − 푐; 푧) (푐 ≠ 2,3,4, … ).

Integral representation of the confluent hypergeometric functions, which are also shown as 1퐹1(푎, 푏; 푧), can be given as

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1 Г(푐) 푀(푎, 푐; 푧) = ∫ e푧푡푡푎−1(1 − 푡)푐−푎−1푑푡 (푎, 푐 ∈ ℝ, 푐 > 푎 > 0). Г(푎)Г(푐 − 푎) 0 [17].

Now, for Eq. (22), we use the transformation as

휑(푧) = 푧−푐⁄2e푧⁄2푢(푧) [푢(푧) = 푧푐⁄2e−푧⁄2휑(푧)]. (23)

So, we can write

푐 푑휑 − −1 푑푢 1 = 푧 2 e푧⁄2 [푧 + (푧 − 푐)푢], (24) 푑푧 푑푧 2 and,

2 푐 2 푑 휑 − −2 푑 푢 푑푢 1 = 푧 2 e푧⁄2 {푧2 + 푧(푧 − 푐) + [(푧 − 푐)2 + 2푐]푢}. (25) 푑푧2 푑푧2 푑푧 4

By substituting (23), (24) and (25) into (22), we have

푐 푑2푢 1 − 푎 2푐 − 푐2 + (− + 2 + ) 푢 = 0. (26) 푑푧2 4 푧 4푧2

After, we can write Eq. (26) as follows

푐 1 푐 − 1 2 푑2푢 1 − 푎 − ( ) + [− + 2 + 4 2 ] 푢 = 0. (27) 푑푧2 4 푧 푧2

By using Theorem (2.9), we have [18]

휇 = 2, 퓅 = 퓆 = 1, 푎0 = ℎ ≠ 0, 푎1 = 0, 푏0 = 푠 ≠ 0, 푏1 = 푡, (28) so that

풫(푧; 1) = ℎ푧, 풫1(푧; 1) = ℎ, (29) and,

풬(푧; 1) = 푠푧 + 푡, 풬1(푧; 1) = 푠. (30)

After, by using Eq. (18), we obtain

Sayfa No: 286 Ö. ÖZTÜRK

푧 풬(휉; 1) ℋ(푧; 1,1) = ∫ 푑휉 = ln[(ℎ푧)푡⁄ℎe푠푧⁄ℎ]. (31) 풫(휉; 1)

3.1. Theorem Let |푓휇(푧)| < ∞ and 푓−휇 ≠ 0. The nonhomogeneous second order linear ordinary differential equation as

푑2휑 푑휑 ℎ푧 + (푠푧 + 휇ℎ + 푡) + 휇푠휑(푧) = 푓(푧) (ℎ ≠ 0, 휇 ∈ ℝ), (32) 푑푧2 푑푧 has a solution as follows

(푡−ℎ)⁄ℎ 푠푧⁄ℎ −푡⁄ℎ −푠푧⁄ℎ 휑(푧) = {[푓−휇(푧)(ℎ푧) e ] (ℎ푧) e } . (33) −1 휇−1

Furthermore, the homogeneous second order linear ordinary differential equation as

푑2휑 푑휑 ℎ푧 + (푠푧 + 휇ℎ + 푡) + 휇푠휑(푧) = 0 (ℎ ≠ 0, 휇 ∈ ℝ), (34) 푑푧2 푑푧 has a solution as follows

휑(푧) = 퐾[(ℎ푧)−푡⁄ℎe−푠푧⁄ℎ] , 휇−1 (35) where 퐾 is an arbitrary constant [18].

Now, by using Theorem (3.1), we set

ℎ = 1, 푠 = −1, 푡 = 푐 − 푎, 휇 = 푎. (36)

So, we obtain the equation as

푑2휑 푑휑 푧 + (푐 − 푧) − 푎휑(푧) = 0. (37) 푑푧2 푑푧

After, we find the solution of Eq. (37) as follows

푎−푐 푧 휑(푧) = 퐾[푧 e ]푎−1. (38)

Finally, we have the solution of Eq. (27) as

푐⁄2 −푧⁄2 푎−푐 푧 푢(푧) = 퐾푧 e [푧 e ]푎−1. (39)

3.2. Example Let 푎 = 3 and 푐 = 1 for Eq. (38) and Eq. (39). So, we obtain

2 푧 휑(푧) = 퐾(푧 e )2, (40) and,

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1⁄2 −푧⁄2 2 푧 푢(푧) = 퐾푧 e (푧 e )2. (41)

By using Eq. (1), we have

푧 1 푑3 (푧2e푧) = ∫ 휏2e휏 푑휏 = e푧(푧2 + 4푧 + 2). (42) 2 Г(1) 푑푧3 0

After, by substituting (42) into (40) and (41), we find the solutions as

휑(푧) = 퐾e푧(푧2 + 4푧 + 2), (43) and,

푢(푧) = 퐾푧1⁄2e푧⁄2(푧2 + 4푧 + 2). (44)

1 3.3. Theorem Let |(푧푎−푐) | < ∞ (푘 ∈ ℤ+ ∪ {0}), 푧 ≠ 0, and | | < 1. The solution of (38) 푘 푧 can be written as follows

1 휑(푧) = 퐾푧푎−푐e푧 퐹 [1 − 푎, 푐 − 푎; ] , (45) 2 0 푧

where 2퐹0 is the Gauss hypergeometric function.

Proof. By means of (8), we have

∞ Γ(푎) 휑(푧) = 퐾 ∑ (푧푎−푐) (e푧) . (46) Γ(푎 − 푘)Γ(푘 + 1) 푘 푎−1−푘 푘=0

By using (9), (11), (12) and (13), we can rewrite the Eq. (46) as follows

∞ Γ(푘 + 1 − 푎) 1 Γ(푘 + 푐 − 푎) 휑(푧) = 퐾 ∑ (−1)푘푧푎−푐−푘 e푧, (−1)푘Γ(1 − 푎) 푘! Γ(푐 − 푎) 푘=0

∞ 1 1 푘 = 퐾푧푎−푐e푧 ∑[1 − 푎] [푐 − 푎] ( ) , 푘 푘 푘! 푧 푘=0

1 = 퐾푧푎−푐e푧 퐹 [1 − 푎, 푐 − 푎; ]. (47) 2 0 푧

4. CONCLUSION

In this paper, we used the differintegral theorems for the confluent hypergeometric equation. We also obtained hypergeometric forms of the fractional solutions. Solutions of the singular equations can be obtained by means of these theorems.

Sayfa No: 288 Ö. ÖZTÜRK

5. ACKNOWLEDGMENT

The author would like to thank the referees for useful and improving comments.

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Fen ve Mühendislik Dergisi Cilt: 18 No: 3 Sayı: 54 Sayfa No: 289

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CV / ÖZGEÇMİŞ

Ökkeş ÖZTÜRK; Yrd.Doç.Dr. (Assist.Prof) Lisans derecesini 2009'da Manisa Celal Bayar Üniversitesi Matematik Bölümü'nden, Yüksek Lisans derecesini 2011'de Elazığ Fırat Üniversitesi Matematik Bölümü'nden, Doktora derecesini 2015 yılında Elazığ Fırat Üniversitesi Matematik Bölümü'nden aldı. Hala Bitlis Eren Üniversitesi Matematik Bölümü'nde öğretim üyesi olarak görev yapmaktadır. Temel çalışma alanları: Kesirli Hesap, Kesirli Hesapta N-Metot, Diferintegral Teoremleri üzerinedir.

He got his bachelors’ degree in the Department of Mathematics at Celal Bayar University, Manisa/Turkey in 2009, his master degree in the Department of Mathematics at Firat University, Elazig/Turkey in 2011, PhD degree in the Department of Mathematics at Firat University, Elazig/Turkey in 2015. He is still an academic member of the Department of Mathematics at Bitlis Eren University. His major areas of interests are: Fractional Calculus, N-Method in the Fractional Calculus, Differintegral Theorems.