An extension of Caputo fractional derivative operator and its applications

Available online at wwwtjnsacom J Nonlinear Sci Appl 9 26, 36 362 Research Article An extension of Caputo fractional derivative operator and its applications İ Onur Kıyma a, Ayşegül Çetinkaya a,, Praveen Agarwal b a Ahi Evran Univ, Dept of Mathematics, 4 Kırşehir, Turkey b Anand International College of Eng, Dept of Mathematics, 332 Jaipur, India Communicated by X-J Yang Abstract In this paper, an extension of Caputo fractional derivative operator is introduced, and the extended fractional derivatives of some elementary functions are calculated At the same time, extensions of some hypergeometric functions and their integral representations are presented by using the extended fractional derivative operator, linear and bilinear generating relations for extended hypergeometric functions are obtained, and Mellin transforms of some extended fractional derivatives are also determined c 26 All rights reserved Keywords: Caputo fractional derivative, hypergeometric functions, generating functions, Mellin transform, integral representations 2 MSC: 26A33, 33C5, 33C2, 33C65 Introduction Special functions are used in the application of mathematics to physical and engineering problems In recent years, many authors considered the several extensions of well known special functions see, for example, [2, 3, 9, 2, 3]; see also the very recent work [8, ] In 994, Chaudhry and Zubair [4], introduced the generalied representation of gamma function In 997, Chaudhry et al [2] presented the following extension of Euler s beta function B p x, y : p t x t y e t t dt, Corresponding author Email addresses: iokiyma@ahievranedutr İ Onur Kıyma, acetinkaya@ahievranedutr Ayşegül Çetinkaya, goyalpraveen2@gmailcom Praveen Agarwal Received 26-2-22

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 362 where Rep >, Rex >, Rey > Recently, Chaudhry et al [3] used B p x, y to extend the hypergeometric functions as F p α, β; γ; : n B p β + n, γ β n, Bβ, γ β where p, Reγ > Reβ > and < The symbol denotes the Pochhammer s symbol defined by Γa + n :, α : Γa Afterwards, in [] Öarslan and Öergin obtained linear and bilinear generating relations for extended hypergeometric functions by defining the extension of the Riemann-Liouville fractional derivative operator as D α,p f : dm d m Dα m dm d m f Γ α + m t α+m e p 2 ftdt where Rep > and m < Reα < m, m N It is obvious that, these extensions given above, coincide with original ones when p The above-mentioned works have largely motivated our present study The principle aim of the paper is to present extension of the Caputo fractional derivative operator and calculating the extended fractional derivatives of some elementary functions In the sequel, extensions of some hypergeometric functions and their integral representations are presented by using the extended fractional derivative operator, linear and bilinear generating relations for extended hypergeometric functions are obtained, and Mellin transforms of some extended fractional derivatives are also determined, 2 Extended hypergeometric functions In this section, we introduce the extensions of Gauss hypergeometric function 2 F, the Appell hypergeometric functions F, F 2 and the Lauricella hypergeometric function FD,p 3 Throughout this paper we assume that Rep > and m N Definition 2 The extended Gauss hypergeometric function is 2F a, b; c; ; p : a n b n B p b m + n, c b + m n b m n Bb m, c b + m n 2 for all < where m < Reb < Rec Definition 22 The extended Appell hypergeometric function F is F a, b, c; d; x, y; p : n,k a n+k b n c k B p a m + n + k, d a + m x n y k a m n+k Ba m, d a + m k! 22 for all x <, y < where m < Rea < Red Definition 23 The extended Appell hypergeometric function F 2 is

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 363 F 2 a, b, c; d, e; x, y; p a n+k b n c k B p b m + n, d b + m x n y k : b m n e k Bb m, d b + m k! n,k n,k n,k a n+k b n c k B p c m + k, e c + m x n y k d n c m k Bc m, e c + m k! [ a n+k b n c k B p b m + n, d b + m b m n c m k Bb m, d b + m B p c m + k, e c + m Bc m, e c + m ] x n y k k! 23 24 25 for all x + y < where m < Reb < Red and m < Rec < Ree Definition 24 The extended Lauricella hypergeometric function F 3 D,p is F 3 D,pa, b, c, d; e; x, y, ; p : n,k,r a n+k+r b n c k d r B p a m + n + k + r, e a + m x n y k r a m n+k+r Ba m, e a + m k! r! 26 for all x + y + < where m < Rea < Ree Note that when p, these functions reduces to well known Gauss hypergeometric function 2 F, Appell functions F, F 2 and Lauricella function F 3 D, respectively 3 Extended Caputo fractional derivative operator The fractional derivative operators has gained considerable popularity and importance during the past few years In literature point of view many fractional derivative operators already proved their importance Very recently, fractional operator, whose derivative has singular kernel introduced by Yang et al [4] Motivated by above work many researchers applied new derivative in certain real world problems see, eg, [5, 7, 5 7] In the sequel, we aim to extend the definition of the classical Caputo fractional derivative operator The classical Caputo fractional derivative is defined by D µ f : m µ dm t dt m ftdt, where m < Reµ < m, m N We refer [6] to the reader for more information about fractional calculus Inspired by the same idea in [], we introduce the Extended Caputo Fractional Derivative as D µ,p f : t m µ e p 2 d m ftdt, 3 dtm where Rep > and m < Reµ < m, m N In the case p, extended Caputo fractional derivative reduces to classical Caputo Fractional derivative, and also when µ m N and p, m, f : f m Now, we begin our investigation by calculating the extended fractional derivatives of some elementary functions Theorem 3 Let m < Reµ < m and Reµ < Reλ then µ,p λ Γλ + B pλ m +, m µ Γλ µ + Bλ m +, m µ λ µ

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 364 Proof With direct calculation, we get µ,p λ t m µ e Γλ + Γλ m + Γλ + Γλ m + λ µ p 2 d m dt m tλ dt t m µ t λ m e u m µ u λ m e Γλ + B pλ m +, m µ Γλ µ + Bλ m +, m µ λ µ Remark 32 Note that, µ,p λ for λ,,, m p 2 p u u The next theorem expresses the extended Caputo fractional derivative of an analytic function Theorem 33 If f is an analytic function on the disk < ρ and has a power series expansion f n a n n, then µ,p f a n µ,p n where m < Reµ < m Proof Using the power series expansion of f, we get µ,p f n t m µ e p 2 n dt a n d m dt m tn dt Since the power series converges uniformly and the integral converges absolutely, then the order of the integration and the summation can be changed So we get, µ,p f a n n a n µ,p n n t m µ e p 2 The proof of the following theorem is obvious from Theorem 3 and 33 d m dt m tn dt Theorem 34 If f is an analytic function on the disk < ρ and has a power series expansion f n a n n, then µ,p λ f where m < Reµ < m < Reλ n a n µ,p λ+n Γλλ µ Γλ µ Γλλ µ Γλ µ n n a n a n du λ n B p λ m + n, m µ λ µ n Bλ m + n, m µ n λ n B p λ m + n, m µ n, λ m n Bλ m, m µ The following theorems will be useful for finding the generating function relations

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 365 Theorem 35 Let m < Reλ µ < m < Reλ, then λ µ,p λ α Γλµ λ n B p λ m + n, µ λ + m n Γµ λ m n Bλ m, µ λ + m for < n Γλ Γµ µ 2F α, λ; µ; ; p Proof If we use the power series expansion of α and 2, we get λ µ,p λ α λ µ,p λ n n λ µ,p λ+n n n Γλ Γµ µ Γλ Γµ µ Γλ + n B p λ m + n, m λ + µ Γµ + n Bλ m + n, m λ + µ µ+n λ n B p λ m + n, m λ + µ n µ n n Bλ m + n, m λ + µ λ n B p λ m + n, µ λ + m n λ m n Bλ m, µ λ + m n Γλ Γµ µ 2F α, λ; µ; ; p Theorem 36 Let m < Reλ µ < m < Reλ, then λ a α b β D λ µ,p for a < and b < Γλ Γµ µ n,k Γλ Γµ µ F λ, α, β; µ; a; b; p λ n+k β k B p λ m + n + k, µ λ + m a n b k λ m n+k Bλ m, µ λ + m k! Proof Using the power series expansion of a α, b β and 22, we get λ µ,p λ a α b β λ µ,p β k a n b k λ+n+k k! n,k n,k Γλ Γµ µ β k k! β k k! n,k a n b k λ µ,p λ+n+k n k a n b k Γλ + n + kb pλ m + n + k, m λ + µ µ+n+k Γλ m + n + kγm λ + µ Γλ Γµ µ F λ, α, β; µ; a; b; p λ n+k β k B p λ m + n + k, m λ + µ a n b k λ m n+k Bλ m, m λ + µ k! 32 33

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 366 Theorem 37 Let m < Reλ µ < m < Reλ, then λ a α b β c γ D λ µ,p Γλ Γµ µ n,k,r λ n+k+r β k γ r B p λ m + n + k + r, µ λ + m a n b k c r λ m n+k+r Bλ m, µ λ + m k! r! Γλ Γµ µ F 3 D,pλ, α, β, γ; µ; a; b; c; p for a <, b < and c < Proof Using the power series expansion of a α, b β, c γ and 26, we get λ a α b β c γ D λ µ,p D λ µ,p n,k,r n,k,r n k r Γλ Γµ µ β k k! β k k! n,k,r γ r r! γ r r! β k k! γ r a n b k c r λ+n+k+r r! a n b k c r λ µ,p λ+n+k+r a n b k c r Γλ + n + k + rb pλ m + n + k + r, m λ + µ µ+n+k+r Γλ m + n + k + rγm λ + µ λ n+k+r β k γ r B p λ m + n + k + r, m λ + µ a n b k c r λ m n+k+r Bλ m, m λ + µ k! r! Γλ Γµ µ F 3 D,pλ, α, β, γ; µ; a; b; c; p Theorem 38 Let m < Reλ µ < m < Reλ and m < Reβ < Reγ, then D λ µ,p λ α x 2F α, β; γ; [ Γλ +k β n λ k Γµ µ β m n λ m k for x + < n k B p β m + n, γ β + m Bβ m, γ β + m Γλ Γµ µ F 2 α, β, λ; γ, µ; x, ; p B p λ m + k, µ λ + m Bλ m, µ λ + m Proof Using the power series expansion of α, F p and 25, we get λ µ,p λ α x 2F α, β; γ; λ µ,p λ α β n B p β m + n, γ β + m x n β m n n Bβ m, γ β + m λ µ,p λ α n β n B p β m + n, γ β + m x n β m n Bβ m, γ β + m n ] x n k k! 34 35

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 367 β n B p β m + n, γ β + m x n β m n n Bβ m, γ β + m Dλ µ,p λ α n [ Γλ +k β n λ k B p β m + n, γ β + m Γµ µ β m n λ m k Bβ m, γ β + m n k Γλ Γµ µ F 2 α, β, λ; γ, µ; x, ; p B p λ m + k, µ λ + m Bλ m, µ λ + m ] x n k k! 4 Generating functions In this section, we use the equalities 32, 33 and 35 for obtaining linear and bilinear generating relations for the extended hypergeometric function 2 F Theorem 4 Let m < Reλ µ < m < Reλ, then n where < min, t Proof Taking the identity in [] and expanding the left hand side, we get 2F α + n, λ; µ; ; pt n t α 2F α, λ; µ; t ; p, 4 n [ t] α t α α t t n α t α α, t when t < If we multiply the both sides with λ and apply the extended Caputo fractional derivative operator λ µ,p, we get λ µ,p t n λ α n λ µ,p t α λ α t n n Since t < and Reλ > Reµ >, it is possible to change the order of the summation and the derivative as α n λ µ,p λ α n t n t α λ µ,p λ α t So we get the result after using Theorem 35 on both sides Theorem 42 Let m < Reλ µ < m < Reλ, then where t < + n 2F β n, λ; µ; ; pt n t α t F β, α, λ; µ; ; t ; p,

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 368 Proof Taking the identity in [] and expanding the left hand side, we get n n [ t] α t α + t α t n t n t α t α, t when t < If we multiply the both sides with λ β and apply the extended Caputo fractional derivative operator λ µ,p, we get λ µ,p λ β n t n λ µ,p t α λ β t α t Since t < t and Reλ > Reµ >, it is possible to change the order of the summation and the derivative as α n λ µ,p λ β+n t n t α λ µ,p λ β t α t n So we get the result after using Theorem 35 and Theorem 36 Theorem 43 Let m < Reβ γ < m < Reβ and m < Reλ < Reµ, then n ut 2F α + n, λ; µ; ; p 2 F n, β; γ; u; p F 2 α, λ, β; µ, γ;, t ; p Proof If we take t ut in 4 and then multiply the both sides with u β, we get n 2F α + n, λ; µ; ; pu β u n t n u β [ ut] α 2F α, λ; µ; ut ; p Applying the fractional derivative D β γ u n α n 2F α + n, λ; µ; ; pdu β γ u β u n t n when <, u t < and t + n to both sides and changing the order we find D β γ u ut t u β [ ut] α2f α, λ; µ; < If we write the equality like α n 2F α + n, λ; µ; ; pdu β γ u β u n t n D β γ u [ u β ut ] α 2F α, λ; µ; t and using Theorem 35 and Theorem 38 we get the desired result ut ; p ut t ; p

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 369 5 Further results and observations In this section, we apply the extended Caputo fractional derivative operator 3 to familiar functions e and 2 F a, b; c; We also obtain the Mellin transforms of some extended Caputo fractional derivatives and we give the integral representations of extended hypergeometric functions Theorem 5 The extended Caputo fractional derivative of f e is for all D µ,p e m µ n n B pm µ, n + Proof Using the power series expansion of e and Theorem 33, we get µ,p e Dµ,p n n Γn + B p n m +, m µ n µ Γn µ + Bn m +, m µ nm Γn + m + B p n +, m µ n+m µ Γn + m µ + Bn +, m µ n + m! n m µ n B pm µ, n + n Theorem 52 The extended Caputo fractional derivative of 2 F a, b; c; is for < µ,p 2F a, b; c; a mb m m µ c m Γ µ + m n a + m n b + m n B p m µ, n + n c + m n µ + m n Bm µ, n + Proof Using the power series expansion of 2 F a, b; c; and making similar calculations, we get µ,p 2 F a, b; c; µ,p a n b n n c n n nm n a n b n µ n c n a n b n c n Γn + B p m µ, n m + Γn µ + Bm µ, n m + n µ a n+m b n+m Γn + m + B p m µ, n + c n n+m n + m! Γn + m µ + Bm µ, n + n+m µ a mb m m µ a + m n b + m n B p m µ, n + n c m Γ µ + m c + m n µ + m n Bm µ, n + n The following two theorems are about the Mellin transforms of extended Caputo fractional derivatives of two functions

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 362 Theorem 53 Let Reλ > m and Res >, then [ M D µ,p λ ] : s Proof Using the definition of Mellin transform we get [ M µ,p λ ] : s p s µ,p λ dp From the equality Γλ + Γs Γλ m + Bm µ + s, λ m + s + λ µ p s Γλ + B p m µ, λ m + Γλ µ + Bm µ, λ m + λ µ dp Γλ + λ µ Γλ µ + Bm µ, λ m + p s B p m µ, λ m + dp b s B p x, ydb ΓsBx + s, y + s, Res >, Rex + s >, Rey + s > in [2, page 2] we get the result Theorem 54 Let Res > and <, then M [ µ,p α :s ] Γs m µ n n Bm µ + s, n + s + µ n n Γn + Proof With using the power series expansion of α and taking λ n in Theorem 53, we get [ M µ,p α ] [ ] : s M µ,p n : s n M [ µ,p n : s] Γs µ Γs m µ nm Bm µ + s, n m + s + n Γn m + Bm µ + s, n + s + +m n n Theorem 55 The following integral representations are valid 2F a, b; c; ; p Bb m, c b + m p t b m t c b+m t t e2 F a, b; b m; t dt, 5 F a, b, c; d; x, y; p Ba m, d a + m t a m t d a+m p t t e F a, b, c; a m; xt, yt dt, 52 F 2 a, b, c; d, e; x, y; p Bb m, d b + m t b m t d b+m p t t e F 2 a, b, c; b m, e; xt, y dt, 53

İ O Kıyma, A Çetinkaya, P Agarwal, J Nonlinear Sci Appl 9 26, 36 362 362 F 2 a, b, c; d, e; x, y; p Bc m, e c + m t c m t e c+m p t t e F 2 a, b, c; d, b m; x, yt dt, 54 F 2 a, b, c; d, e; x, y; p Bb m, e b + mbc m, e c + m t b m u c m t d b+m u e c+m p t t p u u e F 2 a, b, c; b m, c m; xt, yτ dtdu 55 Proof The integral representations 5-55 can be obtained directly by replacing the function B p with its integral representation in 2-25, respectively Acknowledgements This work was partly presented on September 3-7, 22 at st International Eurasian Conference on Mathematical Sciences and Applications IECMSA-22 at Prishtine-KOSOVO This work was supported by Ahi Evran University Scientific Research Projects Coordination Unit Project Number: PYO-FEN444 References [] M Ali Öarslan, E Öergin, Some generating relations for extended hypergeometric functions via generalied fractional derivative operator, Math Comput Modelling, 52 2, 825 833, 3 [2] M A Chaudhry, A Qadir, M Rafique, S M Zubair, Extension of Euler s beta function, J Comput Appl Math, 78 997, 9 32, 5 [3] M A Chaudhry, A Qadir, H M Srivastava, R B Paris, Extended hypergeometric and confluent hypergeometric functions, Appl Math Comput, 59 24, 589 62 [4] M A Chaudhry, S M Zubair, Generalied incomplete gamma functions with applications, J Comput Appl Math, 55 994, 99 23 [5] J H He, A tutorial review on fractal spacetime and fractional calculus, Internat J Theoret Phys, 53 24, 3698 378 3 [6] A A Kilbas, H M Srivastava, J J Trujillo, Theory and applications of fractional differential equations, Elsevier Science B V, Amsterdam, 26 3 [7] F J Liu, Z B Li, S Zhang, H Y Liu, He s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy, Therm Sci, 9 25, 55 59 3 [8] R K Parmar, Some Generating Relations For Generalied Extended Hypergeometric Functions Involving Generalied Fractional Derivative Operator, J Concr Appl Math, 2 24, 27 228 [9] H M Srivastava, P Agarwal, S Jain, Generating functions for the generalied Gauss hypergeometric functions, Appl Math Comput, 247 24, 348 352 [] H M Srivastava, A Çetinkaya, İ O Kıyma, A certain generalied Pochhammer symbol and its applications to hypergeometric functions, Appl Math Comput, 226 24, 484 49 [] H M Srivastava, H L Manocha, A treatise on generating functions, Halsted Press, New York, 984 4, 4 [2] E Öergin, Some properties of hypergeometric functions, PhD Thesis, Eastern Mediterranean University, North Cyprus, Turkey, 2 [3] E Öergin, M A Öarslan, A Altın, Extension of gamma, beta and hypergeometric functions, J Comput Appl Math, 235 2, 46 46 [4] X J Yang, D Baleanu, H M Srivastava, Local fractional integral transforms and their applications, Academic Press, Amsterdam, 26 3 [5] X J Yang, H M Srivastava, An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives, Commun Nonlinear Sci Numer Simul, 29 25, 499 54 3 [6] X J Yang, H M Srivastava, J A Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, arxiv preprint, 26 [7] X J Yang, J A Tenreiro Machado, A new insight into complexity from the local fractional calculus view point: modelling growths of populations, Math Meth Appl Sci, 25 25, 6 pages 3