Laplace Operator with Caputo-Type Marichev–Saigo–Maeda Fractional Differential Operator of Extended Mittag- Leffler Function

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Hindawi Discrete Dynamics in Nature and Society Volume 2021, Article ID 5587616, 11 pages https://doi.org/10.1155/2021/5587616

Research Article Laplace Operator with Caputo-Type Marichev–Saigo–Maeda Fractional Differential Operator of Extended Mittag- Leffler Function

Adnan Khan,1 Tayyaba Manzoor,1 Hafte Amsalu Kahsay ,2 and Kahsay Godifey Wubneh 2

1National College of Business Administration & Economics, Lahore, Pakistan 2Wollo University, College of Natural Science, Department of Mathematics, Dessie, Ethiopia

Correspondence should be addressed to Hafte Amsalu Kahsay; [email protected]

Received 6 January 2021; Revised 26 February 2021; Accepted 1 April 2021; Published 19 April 2021

Academic Editor: Xiaohua Ding

Copyright © 2021 Adnan Khan et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the Laplace operator is used with Caputo-Type Marichev–Saigo–Maeda (MSM) fractional differentiation of the extended Mittag-Leffler function in terms of the Laplace function. Further in this paper, some corollaries and consequences are shown which are the special cases of our main findings. We apply the Laplace operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. We also apply the Laplace operator on the right-sided MSM fractional differential operator with the Mittag-Leffler function and the left-sided MSM fractional differential operator with the Mittag-Leffler function.

1. Introduction modifying it by replacing the power function kernel with other kernel functions. (ey demonstrated, under some Fractional calculus is a fast-growing field of mathematics assumptions, how all of these modifications can be con- that shows the relations of fractional-order derivatives and sidered as special cases of a single, unifying model of integrals. Fractional calculus is an effective subject to study fractional calculus. (ey provided a fundamental connection many complex real-world systems. In recent years, many with classical fractional calculus by writing these general researchers have calculated the properties, applications, and fractional operators in terms of the original Rie- extensions of fractional integral and differential operators mann–Liouville fractional integral operator. involving various special functions. Fernandez et al. [2] considered an integral transform Integral and differential operators in fractional calculus introduced by Prabhakar, involving generalized multipa- have become a research subject in recent decades due to the rameter Mittag-Leffler functions, which can be used to in- ability to have arbitrary order. Special functions are the troduce and investigate several different models of fractional functions that have improper integrals or series. Some of the calculus. (ey derived a new series expression for this well-known functions are the gamma function, beta func- transform, in terms of classical Riemann–Liouville fractional tion, and hypergeometric function. integrals, and used it to obtain or verify series formulas in Many researchers established compositions of new frac- various specific cases corresponding to different fractional tional derivative formulas called Marichev–Saigo–Maeda calculus models. Caputo-type fractional operators on well-known functions Srivastava et al. [3] considered the well-known Mittag- like the Mittag-Leffler function. Leffler functions of one, two, and three parameters and Fernandez et al. [1] proposed the definitions for frac- established some new connections between them using tional derivatives and integrals, starting from the classical fractional calculus. In particular, they expressed the three- Riemann–Liouville formula and its generalizations and parameter Mittag-Leffler function as a fractional derivative 2 Discrete Dynamics in Nature and Society of the two-parameter Mittag-Leffler function, which is, in nowadays. (ey considered a fractional-order epidemic turn, a fractional integral of the one-parameter Mittag- model which describes the dynamics of COVID-19 under a Leffler function. Hence, they derived an integral expression non-singular kernel type of fractional derivative. An attempt for the three-parameter one in terms of the one-parameter is made to discuss the existence of the model using the fixed one. point theorem of Banach and Krasnoselskii. Khan et al. [4] studied the fractional-order model of Manzoor et al. [10] used the beta operator with Caputo HIV/AIDS involving the Liouville–Caputo and Atanga- (MSM) fractional differentiation of the extended Mittag- na–Baleanu–Caputo derivatives. (e generalised HIV/AIDS Leffler function in terms of beta function. (ey applied the model allows and shows that certain infected individuals beta operator on the right-sided MSM fractional differential switch from symptomatic to asymptomatic phases. Special operator and on the left-sided MSM fractional differential iterative solutions were obtained by the use of Laplace and operator. (ey also applied the beta operator on the right- Sumudu transform. sided MSM fractional differential operator with the Mittag- Khan [5] established the existence of positive solutions Leffler function and the left-sided MSM fractional differ- (EPS) and the Hyers–Ulam (HU) stability of a general class ential operator with the Mittag-Leffler function. of nonlinear Atangana–Baleanu–Caputo (ABC) fractional Kilbas et al. [11] worked on the composition of Rie- differential equations (FDEs) with singularity and nonlinear mann–Liouville fractional integration and differential op- p−Laplacian operator in Banach’s space. erators. Rao et al. [12] introduced the result that fractional Khan et al. [6] are interested in using the Atanga- integration and fractional differentiation are interchanged. na–Baleanu fractional differential form to analyse the HIV- Agarwal and Jain [13] developed fractional calculus formula TB co-infected model. (e model is studied for the existence, of polynomial using the series expansion method. Choi and uniqueness of solution, Hyers–Ulam (HU) stability, and Agarwal [14] aimed to find confident integral transforms numerical simulations with the assumption of specific and fractional integral formula for the generalized hyper- parameters. geometric function. Agarwal and Choi [15] proved certain Ahmad et al. [7] presented the mathematical model with image formulas of various fractional integral operators in- different compartments for the transmission dynamics of volving Gauss hypergeometric function. Further, it is coronavirus-19 disease (COVID-19) under the fractional- expressed in terms of Hadamard product. order derivative. Some results regarding the existence of at Nadir and Khan [16] applied Caputo-type MSM frac- least one solution through fixed point results have been tional differentiation on the Mittag-Leffler function. Nadir derived. (en, for the concerned approximate solution, the and Khan [17, 18] used fractional integral operator asso- modified Euler method for fractional-order differential ciated with the extended Mittag-Leffler function. equations (FODEs) is utilized. Nadir et al. [19] studied the extended versions of the Shah et al. [8] studied a compartmental mathematical generalized Mittag-Leffler function. Nadir and Khan [20] model for the transmission dynamics of the novel corona- applied Weyl fractional calculus operators on the extended virus-19 under Caputo fractional-order derivative. By using Mittag-Leffler function. Mondal and Nisar [21] applied the the fixed point theory of Schauder and Banach, they Marichev–Saigo–Maeda operator on the Bessel function. established some necessary conditions for the existence of at Nadir and Khan [22] applied the Marichev–Saigo–Maeda least one solution to model under investigation and its differential operator and generalized incomplete hyper- uniqueness. After the existence, a general numerical algo- geometric functions. Maitama and Zhao [23] worked on a rithm based on the Haar collocation method is established to new integral transform called Shehu transform, a general- compute the approximate solution of the model. ization of Sumudu and Laplace transform, for solving dif- Sher et al. [9] studied the novel coronavirus (2019-nCoV ferential equations. or COVID-19) which is a threat to the whole world Srivastava et al. [24] defined a function

⎧⎪ |x < R| ⎫⎪ ⎪ ⎪ ⎪ ⎜⎛ ⎟⎞ ⎪ ⎪ n ⎜ ⎟ ⎪ ⎪ ∞ z ⎜ ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ � kn ⎜ 0 < R < ∞ ⎟ ⎪ ⎪ n ! ⎜ ⎟ ⎪ ⎪ n�0 ⎝⎜ ⎠⎟ ⎪ ⎨⎪ ⎬⎪ k Θ��n �n∈N ; x � ≔ ⎪ k0 ≔ 1 ⎪, (1) 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R(x) ⟶ ∞ ⎪ ⎪ 1 ⎜⎛ ⎟⎞ ⎪ ⎪ m xϖ exp(x) 1 + 0 � �� ⎝⎜ ⎠⎟ ⎪ ⎩⎪ 0 2 ⎭⎪ m0 > 0; ϖ ∈ C Discrete Dynamics in Nature and Society 3 where Θ(�k � ; x) is considered to be analytical with n n∈N0 2. The Confluent Hypergeometric Function |x| < R, 0 < R < ∞, �k � is a sequence of Tay- n n∈N0 lor–Maclaurin coefficient, and m0 and ϖ are constants and (e confluent hypergeometric function by Rainville is de- depend upon the bounded sequence �k � . fined as 2F1(a, b; c; x) which is represented by hyper- n n∈N0 (e series geometric series. ∞ (a) (b) xm �{}kn ;c � m m n ∈ N0 F (a, b; c; x) � � . ·( ) Eε,μ (x, p) 2 1 8 m�0 (c)m m! �{}kn ;c � (2) ∞ B n ∈ N0 (c + k, 1 − c: ρ) xk � � ρ , B(c, − c) ( k + ) k�0 1 Γ ε μ 3. The Hadamard Product of the Power Series x, μ, c ∈ C; R(ε) > 0 ∞ m where � � is known as the ex- As indicated in Pohlen [28], let g(z) � �m�0 xmz and R(μ) > 0, R(c) > 1: ρ ≥ 0 ∞ m h(z) � �m�0 ymz be two power series; then, the Hadamard tension of the Mittag-Leffler function. It was defined by product of power series is defined as Parmar [25]. ∞ Mittag-Leffler functions with special cases are given m (g ∗ h)(z) � � xmymz �(h · g)(z), (|z| < R), (9) below. m�0 (i) When Kn � ((ρ)n/(σ)n), then the extended form of where (2) takes the form � � � � � � � x y � � x � � y � (ρ,σ) � m m � � m � � m � ∞ k R � lim � ��lim � � ��lim � � � � Rg · Rh, (ρ,σ): c Bρ (c + k, 1 − c: ρ) x m⟶∞�x y � k⟶∞�x � m⟶∞�y � E (x; ρ) � � , m+1 m+1 m+1 m+1 ε,μ B(c, − c) ( k + ) k�0 1 Γ ε μ (10) (3) where Rg and Rh are radii of convergence of the above series under the condition g(z) and h(z), respectively. (erefore, in general, it is to be x, μ, c ∈ C; R(ε) > 0, R(σ) > 0 noted that if one power series is an analytical function, then � �. (4) the series of Hadamard products is also the same like an R(ε) > 0, R(μ) > 0, R(c) > 1; ρ ≥ 0 analytical function.

(ii) If we select a bounded sequence Kn � 1, then (2) 4. Laplace Transform reduces to the deﬁnition of Ozarslan¨ and Yilmaz [26]. (e Laplace transform of the function f(x) on an interval ∞ k [0, ∞) by Sneddon [29] is deﬁned as c B(c + k, 1 − c: ρ) x E (x; ρ) � � , ∞ ε,μ B(c, − c) ( k + ) − lx k�0 1 Γ ε μ L[f(x); l] � � e f(x)dx, (11) (5) 0 x, μ, c ∈ C; R(ε) > 0 where l ∈ C and x ≥ 0. · ⎛⎝ ⎞⎠. R( ) , R(c) : μ > 0 > 1 ρ ≥ 0 5. Appell Function

(iii) Another special case of (2) is when Kn � 1 and Appell function by Rainville [30] of first kind F3 is basically ρ � 0; then, (2) reduces to Prabhakar’s function [27] two-variable hypergeometric function defined as of three parameters. ∞ (ω) ω′ � (υ) υ′ � xm yn F ω, ω′, υ, υ′, η; x, y� � � m n m n ∞ c �xk 3 (η) m! n! Ec (x; ρ) � � k , m,n m+n ε,μ Γ(εk + μ)k! k�0 (6) ∞ ′, ′ m (ω) (υ) ω υ x � � m m F ⎣⎢⎡ ; y ⎦⎥⎤ . (ε, μ, c ∈ c; R(ε) > 0, R(μ) > 0). (η) 2 1 m! m�0 m η + m (iv) If we set ε � μ � 1, then our expression for (12) ({}kn ;c) n ∈ N0 (ρ,σ): c c Eε,μ , Eε,μ , and Eε,μ reduces to the extended confluent hypergeometric functions: 6. The Left-Sided MSM Fractional

�{}kn ;c � �{}kn � n ∈ N0 n ∈ N0 Differential Operator Eε,μ (x; p) � Φp (c; 1; x), ((p.q);c) (p,q) (7) (e left-sided MSM fractional diﬀerential operator con- Eε,μ (x; p) � Φp (c; 1; x), taining Appell F3 function in their kernel by Saigo and c , , , , , C E1,1(x; p) � ΦΡ(c; 1; x). Maeda [31] is deﬁned as follows: let α α1 ω ω1 μ ρ ∈ and x > 0; then, 4 Discrete Dynamics in Nature and Society

, , , , − ,− ,− ,− ,− α α1 ω ω1 μ α α1 ω ω1 μ Lemma 2. Let ω, λ, β, ρ ∈ c, x > 0 such that R(ω) > 0; then, D0+ f �(x) � I0+ f�(x) Γ(1 − ρ − λ)Γ(1 − ρ + ω + β) n (13) ω,λ,βtρ− 1 ρ+λ+1, d ,− ,− ,− +n,+ +n �D− �(x) � x � �Iα α1 ω ω1 μ f�(x), Γ(1 − ρ + β − λ)Γ(1 − ρ) dxn 0+ (17) where R(μ) > 0 and n � [R(−μ) + 1]. where 7. The Right-Sided MSM Fractional (R(ρ − σk) < 1 + min�R(−λ − n), R(β + ω)� and [R(ω)] + 1). Differential Operator (18) (e right-sided MSM fractional diﬀerential operator con- taining Appell function F3 in their kernel by Saigo and Maeda [31] is deﬁned as follows. Lemma 3. Let β, β′, ε, ε′, η, ρ ∈ C and m � [R(η)] + 1, Let α, α , ω, ω , μ, ρ ∈ C and x > 0. (en, o, R(−β + ε) 1 1 R(ρ) − m > max� �; then, the image , , , , − ,− ,− ,− ,− R(−β − β′ − ε′ + η) Dα α1 ω ω1 μf �(x) � I α α1 ω ω1 μf�(x) 0− 0− will be n n d α,−α ,−ω+n,−ω ,−μ+n β,β′,ε,ε′,η ρ− 1 �(−1) �I 1 1 f�(x), �D+ t �(x) dxn x,∞ (14) Γ(ρ)Γ(ρ − ε + β)Γ β + β′ + ε′ − η + ρ� � xβ+β′− η+ρ− 1. where R(μ) > 0 and n � [R(−μ) + 1]. Γ(−ε + ρ)Γ β + ε′ − η + ρ�Γ β + β ′ − η + ρ� (19) Lemma 1. Let ω, λ, β, ρ ∈ c, x > 0 such that R(ω) > 0; then, Γ(ρ)Γ(ρ + β + ω + λ) �Dω,λ,βtρ− 1 �(x) � xρ+λ+1, (15) + Γ(ρ + β)Γ(ρ + λ) Lemma 4. Let β, β′, ε, ε′, η, ρ ∈ C and m � [R(η)] + 1, R(−ε′), R(β′ + ε − η) where R(ρ) + m > max� �; then, R(β + β′ − ε′ − η) + m (R(ρ) > − min� 0, R(ω + λ + β)�). (16)

β,β′,ε,ε′,η ρ− 1 �D− t �(x) (20) Γ1 + ε′ − ρ �Γ 1 − β − β′ + η − ρ�Γ 1 − β′ − ε + η − ρ� � xβ+β′− η+ρ− 1. Γ(1 − ρ)Γ 1 − β′ + ε′ − ρ�Γ 1 − β − β′ − ε + η − ρ�

8. The Left-Sided MSM Fractional Differential Operator with Mittag-Leffler Function

Theorem 1. Let ω, λ, β, ρ ∈ C R > 0 be such that (R(ρ+ σk) > − min� 0, R(ω + λ + β)�); then, the following result holds.

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ L⎩ u D+ t Eε,μ x(tu) � (z)⎭ (21) ρ+λ− 1 Z �{}kn ;c � △ n ∈ N0 σ ⎣⎡⎢ σ ⎦⎤⎥ � l Eε,μ x(z/s) � ∗ 4Ψ2 ; x(z/s) � , s △′ Discrete Dynamics in Nature and Society 5 where Proof △ ��l, σ), (ρ, σ), (ρ + β + ω + λ, σ)(1, σ)�, (22) △′ �� (ρ + β, σ), (ρ + λ, σ)�.

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ L⎩ u D+ t Eε,μ x(tu) � (z)⎭ (23) ∞ ⎧⎨ �{}kn ;c � ⎫⎬ − su l− 1 ⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ � � e u ⎩ D+ t Eε,μ x(tu) � (z)⎭du. 0

By using deﬁnition (11) and Lemma (15) and changing ρ by ρ + σk, we get

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ L⎩ u D+ t Eε,μ x(tu) � (z)⎭

�{}kn � ∞ B n ∈ N0 (r + k, 1 − c; p) xk � zρ+σk+λ− 1 � p k�o B(c, 1 − c) Γ(εk + μ) Γ(ρ + σk)Γ(ρ + σk + β + ω + λ) ∞ × � e− suul− 1+σkdu, (24) Γ(ρ + σk + β)Γ(ρ + σk + λ) o

�{}kn � zρ+σk+λ− 1 ∞ B n ∈ N0 (r + k, 1 − c; p) xk � � p l+σk s k�o B(c, 1 − c) Γ(εk + μ) Γ(l + σk)Γ(ρ + σk)Γ(ρ + σk + β + ω + λ) × . Γ(ρ + σk + β)Γ(ρ + σk + λ)

By using Hadamard product which is given in (9), we get Select a bounded sequence kn � 1in equation (25) and then proceed (26). ⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ L⎩ u D+ t Eε,μ x(tu) � (z)⎭ Corollary 2. Let ω, λ, β, ρ ∈ CR > 0 be such that (R(ρ+ (25) σk) > − min� 0, R(ω + λ + β)�); under the stated conditions, ρ+λ− 1 △ Z �{}kn ;c � n ∈ N0 σ ⎣⎢⎡ σ ⎦⎥⎤ the right-sided Caputo fractional differential operator of the � l Eε,μ x(z/s) � ∗ 4Ψ2 ; x(z/s) � . s △′ extended Mittag-Leffler function is defined by □ ⎧⎨ �{}kn � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 σ ⎞⎠⎞⎠ L⎩ u D+ t Φp c; 1; x(tu) � (z)⎭ Corollary 1. Let ω, λ, β, ρ ∈ CR > 0 such that (R(ρ+ σk) > − min� 0, R(ω + λ + β)�); under the stated conditions, ρ+λ− 1 △ Z �{}kn � the right-sided Caputo fractional differential operator of the n ∈ N0 σ ⎢ σ � Φ c ; 1; x(z/s) � ∗ Ψ ⎣⎢⎡ ; x(z/s) �⎦⎥⎤, extended Mittag-Leffler function is defined by l p 4 2 s △′ l− 1 ω,λ,β ρ− 1 c σ (28) L� u �D+ �t Eε,μ �x(tu) 􏰀(z)� where + − (26) △ � �l, σ), (ρ, σ), (ρ + β + ω + λ, σ)(1, σ)�, Zρ β 1 △ ( ) c σ ⎣⎢⎡ σ ⎦⎥⎤ 29 � l Eε,μx(z/s) � ∗ 4Ψ2 ; x(z/s) � , △′ �� (ρ + β, σ), (ρ + λ, σ)�. s △′ where If we select ξ � μ � 1, then extension of Mittag-Leffler function can be expressed in terms of the extended confluent △ �� (l, σ), (ρ, σ), (ρ + β + ω + λ, σ)(1, σ)�, (27) hypergeometric functions. △′ �� (ρ + β, σ), (ρ + λ, σ)�. Theorem 2. Let 6 Discrete Dynamics in Nature and Society

β, β′, ε, ε′, η, ρ ∈ C, where m �[R(η)] + 1, △ �� (l, σ), (ρ, σ)(β − ε + ρ, σ), β + β′ + ε′ − η + ρ, σ�, (1, σ)�, (30) o, R(−β + ε) △′ �� (−ε + ρ, σ), β + β′ − η + ρ, σ�, β + ε′ − η + ρ, σ��. R(ρ) − m > max� �. R −β − β′ − ε′ + η� (32) ?en, Proof ⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ L u D+ t Eε,μ x(tu) � (u) ⎩ ⎭ ⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ L⎩ u D+ t Eε,μ x(tu) � (u)⎭.

β+β′− η+ρ− 1 △ Z �{}kn ;c � n ∈ N0 σ ⎢⎡ σ ⎥⎤ (33) � Eε,μ x(z/s) �∗ Ψ ⎣⎢ ; x(z/s) �⎦⎥, sl 5 3 △′ By using deﬁnition of (11) and Lemma (19) and changing (31) ρ by ρ + σk, we get

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ σ ⎞⎠ L⎩ u D+ t Eε,μ x(tu) � (u)⎭

�{}kn � ∞ B n ∈ N0 (r + k, 1 − c; p) xk � zβ+β′− η+ρ+σk− 1 � p k�o B(c, 1 − c) Γ(εk + μ)

Γ(ρ + σk)Γ(ρ − ε + β + σk)Γ β + β′ + ε′ − η + ρ + σk� × Γ(−ε + ρ + σk)Γ β + β′ − η + ρ + σk�Γ β + ε′ − η + ρ + σk� (34) ∞ × � e− suul− 1+σkdu, o

�{}kn � Zβ+β′− η+ρ+σk− 1 ∞ B n ∈ N0 (r + k, 1 − c; p) xk � � p �� l+σk s k�o B(c, 1 − c) (εk + μ)

Γ(l + σk)Γ(ρ + σk)(Γρ + σk − ε + β)Γ β + β′ + ε′ − η + ρ + σk� × . Γ(−ε + ρ + σk)Γ β + ε′ − η + ρ + σk�Γ β + β′ − η + ρ + σk�

L� ul− 1�Dβ,β′,ε,ε′,η�tρ− 1Ec �x(tu)σ 􏰂(u)� By using Hadamard product which is given in (9), we get + ε,μ

⎧⎨ �{}kn ;c � ⎫⎬ + ′− + − L ul− 1⎛⎝Dβ,β′,ε,ε′,η⎛⎝tρ− 1E n ∈ N0 ⎞⎠x(tu)σ �⎞⎠(u) Zβ β η ρ 1 △ ⎩ + ε,μ ⎭ c σ ⎣⎢⎡ σ ⎦⎥⎤ � l Eε,μx(z/s) � ∗ 5Ψ3 ; x(z/s) � , s △′ β+β′− η+ρ− 1 △ Z �{}kn ;c � (36) n ∈ N0 σ ⎣⎢⎡ σ ⎦⎥⎤ � l Eε,μ x(z/s) � ∗ 5Ψ3 ; x(z/s) � . s △′ where (35) □ △ �� (l, σ), (ρ, σ)(β − ε + ρ, σ), β + β′ + ε′ − η + ρ−, σ�, (1, σ)�, △′ �� (−ε + ρ−, σ), β + β′ − η + ρ, σ�, β + ε′ − η + ρ, σ��. Corollary 3. Let the parameters β, β′, ε, ε′, η, ρ ∈ C and (37) m � [R(η)] + 1, and under the stated conditions, the left- sided Caputo fractional differential operator of the extended Select a bounded sequence kn � 1 in equation (35) and Mittag-Leffler function is defined by then proceed (36). Discrete Dynamics in Nature and Society 7

Corollary 4. Let the parameters β, β′, ε, ε′, η, ρ ∈ C and sided Caputo fractional differential operator of the extended m � [R(η)] + 1, and under the stated conditions, the left- Mittag-Leffler function is defined by

⎧⎨ �{}kn � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 σ ⎞⎠⎞⎠ L⎩ u D+ t Φp c; 1; x(tu) � (u)⎭ (38) β+β′− η+ρ− 1 Z �{}kn � △ n ∈ N0 σ σ � Φp c; 1; x(z/s) � ∗ Ψ3� ; x(z/s) ��, sl 5 △′ where ⎧⎨ �{}kn ;c � ⎫⎬ L ul− 1⎛⎝Dω,λ,β⎛⎝tρ− 1E n ∈ N0 ⎞⎠ x(t u)− σ ⎞⎠(z) △ �� (l, σ), (ρ, σ)(β − ε + ρ − m, σ), ⎩ − ε,μ / � ⎭ · β + β′ + ε′ − η + ρ − m, σ�, (1, σ)�, (39) △′ �� (−ε + ρ − m, σ), β + β′ − η + ρ, σ�, ρ+λ− 1 △ Z �{}kn ;c � n ∈ N0 σ ⎣⎢⎡ − σ ⎦⎥⎤ · β + ε′ − η + ρ − m, σ��. � l Eε,μ x(zs) � ∗ 4Ψ2 ; x(zs) � , s △′ If we select ξ � μ � 1, then extension of the Mittag-Leﬄer (40) function can be expressed in terms of the extended conﬂuent hypergeometric functions. where △ ��l, σ), (1 − ρ − λ, σ), (1 − ρ + ω + β, σ)(1, σ)�, (41) 9. The Right-Sided MSM Fractional Differential △′ �� (1 − ρ + β − λ, σ), (1 − ρ)�. Operator with Mittag-Leffler Function

Theorem 3. ω, λ, β, ρ ∈ CR > 0 where (R(ρ − σk) < 1+ Proof. min�R(−λ − n), R(β + ω)� and n � [R(ω)] + 1; then, the following result holds true:

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ L⎩ u D− t Eε,μ x(t/u) � (z)⎭ (42) ∞ ⎧⎨ �{}kn ;c � ⎫⎬ − su l− 1 ⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ � � e u ⎩ D− t Eε,μ x(t/u) � (z)⎭dz. 0

By using deﬁnition of (11) and Lemma (17) and changing ρ by ρ − σk, we get

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ L⎩ u D− t Eε,μ x(t/u) � (z)⎭

�{}kn � ∞ B n ∈ N0 (r + k, 1 − c; p) xk � zρ− σk+λ− 1 � p k�o B(c, 1 − c) Γ(εk + μ) Γ(1 − ρ + σk − λ)Γ(1 − ρ + σk + ω + β) ∞ × × � e− suul− 1+σkdu, (43) Γ(1 − ρ + σk + β − λ)Γ(1 − ρ + σk) o

�{}kn � zρ− σk+λ− 1 ∞ B n ∈ N0 (r + k, 1 − c; p) xk � � p l+σk s k�o B(c, 1 − c) Γ(εk + μ) Γ(l + σk)Γ(1 − ρ + σk − λ)Γ(1 − ρ + σk + ω + β) × . Γ(1 − ρ + σk + β − λ)Γ(1 − ρ + σk) 8 Discrete Dynamics in Nature and Society

By using Hadamard product which is given in (9), we get where

⎧⎨ �{}kn ;c � ⎫⎬ △ ��l, σ), (1 − ρ − λ, σ), (1 − ρ + ω + β, σ)(1, σ)�, l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ L⎩ u D− t Eε,μ x(t/u) � (z)⎭ (48) △′ �� (1 − ρ + β − λ, σ), (1 − ρ)�.

ρ+λ− 1 Z �{}kn ;c � n ∈ N0 − σ ( ) If we select ξ � μ � 1, then extension of the Mittag- � Eε,μ x(zs) � 44 sl Leffler function can be expressed in terms of the extended △ confluent hypergeometric functions. ⎡⎣ − σ ⎤⎦ ∗ 4Ψ2 ; x(zs) � . △′ □ Theorem 4. Let β, β′, ε, ε′, η, ρ ∈ C and m � [R(η)] + 1, R(−ε′), R(ω′ + ε − η) Corollary 5. Let ω, λ, β, ρ ∈ CR > 0 where (R(ρ − σk) < 1 + R(ρ)+ m > max� �. R(β + β′ − ε′ − η) + m min�R(−λ − n), R(β + ω)� and [R(ω)] + 1); under the stated conditions, the left-sided Caputo fractional differential ⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ operator of the extended Mittag-Leffler function is defined by L⎩ u D− t Eε,μ x(t/u) � (u)⎭ l− 1 ω,λ,β ρ− 1 c − σ L� u �D− �t Eε,μ �x(t/u) 􏰀(z)� ρ+λ− 1 △ + − Z �{}kn ;c � ρ λ 1 n ∈ N0 − σ ⎢ − σ ⎥ Z c − ⎣⎢⎡ ⎦⎥⎤ σ � l Eε,μ x(zs) � ∗ 4Ψ2 ; x(zs) � , � l Eε,μx(zs) � s s (45) △′ (49) △ − σ ∗ 4Ψ2� ; x(zs) ��, △′ where where △ �� (l, σ), 1 − β′ − ε + η − ρ, σ�, 1 + ε′ − ρ, σ �, · 1 − β − β′ + η − ρ, σ�(1, σ)�, △ ��l, σ), (1 − ρ + σ − λ), (1 − ρ + ω + β, σ)(1, σ)�, (50) (46) △′ �� (1 − ρ, σ), 1 − β′ + ε′ − ρ, σ�, △′ �� (1 − ρ + β − λ, σ), (1 − ρ)�. · 1 − β − β′ − ε + η − ρ, σ��.

Select a bounded sequence kn � 1 in equation (44) and then proceed (45). Proof. Corollary 6. Let ω, λ, β, ρ ∈ CR > 0 where (R(ρ − σk) < 1 + min�R(−λ − n), R(β + ω)� and [R(ω)] + 1); under the stated conditions, the left-sided Caputo fractional differential operator of the extended Mittag-Leffler function is defined by

⎧⎨ �{}kn � ⎫⎬ l− 1⎛⎝ ω,λ,β⎛⎝ ρ− 1 n ∈ N0 − σ ⎞⎠⎞⎠ L⎩ u D− t Φp c; 1; x(t/u) � (z)⎭

ρ+λ+1 △ Z �{}kn � n ∈ N0 − σ ⎣⎢⎡ − σ ⎦⎥⎤ � l Φp c; 1; x(zs) � ∗ 4Ψ2 ; x(zs) � , s △′ (47)

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ L⎩ u D− t Eε,μ x(t/u) � (u)⎭ (51) ∞ ⎧⎨ �{}kn ;c � ⎫⎬ − su l− 1 ⎛⎝ β,β′,ε,ε′+η⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ � � e u ⎩ D− t Eε,μ x(t/u) � (z)⎭du. 0 Discrete Dynamics in Nature and Society 9

By using deﬁnition (11) and Lemma (17) and changing ρ by ρ − σk, we get

⎧⎨ �{}kn ;c � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 ⎞⎠ − σ ⎞⎠ L⎩ u D− t Eε,μ x(t/u) � (u)⎭

�{}kn � ∞ B n ∈ N0 (r + k, 1 − c; p) xk � zβ+β′− η+ρ− σk− 1 � p k�o B(c, 1 − c) Γ(εk + μ) Γ 1 + ε′ − ρ + σk� Γ 1 − β − β′ + η − ρ + σk�Γ 1 − β′ − ε + η − ρ + σk� × Γ(1 − ρ + σk)Γ 1 − β′ + ε′ − ρ + σk�Γ 1 − β − β′ − ε + η − ρ + σk� ∞ � e− suul− 1+σkdz, o

�{}kn � � zβ+β′− η+ρ− σk− 1 ∞ B n ∈ N0 (r + k, 1 − c; p) xk · � p l+σk s k�o B(c, 1 − c) Γ(εk + μ) Γ(l + σk)Γ 1 + ε′ − ρ + σk� Γ1 − β − β′ + η − ρ + σk�Γ 1 − β′ − ε + η − ρ + σk� × . (52) Γ(1 − ρ + σk)Γ 1 − β′ + ε′ − ρ + σk�Γ 1 − β − β′ − ε + η − ρ + σk�

By using Hadamard product which is given in (9), we get sided Caputo fractional differential operator of the extended Mittag-Leffler function is defined by ⎧⎨ �{}kn ;c � ⎫⎬ L ul− 1⎛⎝Dβ,β′,ε,ε′,η⎛⎝tρ− 1E n ∈ N0 ⎞⎠x(t u)− σ �⎞⎠(u) ⎩ − ε,μ / ⎭ l− 1 β,β′,ε,ε′,η ρ− 1 c − σ L� u �D− �t Eε,μ �x(t/u) 􏰂(z)� (53) ρ+λ− 1 △ Z �{}kn ;c � n ∈ N0 − σ ⎢⎡ − σ ⎥⎤ + ′− + − � E , x(zs) � ∗ Ψ ⎣⎢ ; x(zs) �⎦⎥. β β η ρ 1 △ l ε μ 4 2 Z c − s ′ σ ⎣⎢⎡ σ ⎦⎥⎤ △ � l Eε,μx(z/s) � ∗ 5Ψ3 ; x(zs) � , □ s △′ (54) Corollary 7. Let the parameters β, β′, ε, ε′, η, ρ ∈ C and m � [R(η)] + 1, and under the stated conditions, the right- where

△ � � l, σ), 1 − β′ − ε + η − ρ, σ�, 1 + ε′ − ρ, σ � 1 − β − β′ + η − ρ, σ�(1, σ��, (55) △′ �� (1 − ρ, σ), 1 − β′ + ε′ − ρ, σ�, 1 − β − β′ − ε + η − ρ, σ��.

Select a bounded sequence kn � 1 in equation (53) and Corollary 8. Let the parameters β, β′, ε, ε′, η, ρ ∈ C and then proceed (54). m � [R(η)] + 1, and under the stated conditions, the right- sided Caputo fractional differential operator of the extended Mittag-Leffler function is defined by

⎧⎨ �{}kn � ⎫⎬ l− 1⎛⎝ β,β′,ε,ε′,η⎛⎝ ρ− 1 n ∈ N0 − σ ⎞⎠⎞⎠ L⎩ u D− t Φp c; 1; x(t/u) � (z)⎭ (56) β+β′− η+ρ− 1 △ Z �{}kn � n ∈ N0 − σ ⎣⎢⎡ − σ ⎦⎥⎤ � l Φp c; 1; x(sz) � ∗ 5Ψ3 ; x(zs) � , s △′ 10 Discrete Dynamics in Nature and Society where

△ �� (l, σ), 1 − β − β′ + η − ρ, σ�, 1 + ε′ − ρ, σ �, 1 − β′ − ε + η − ρ, σ�(1, σ)�, (57) △′ �� (1 − ρ, σ), 1 − β′ + ε′ − ρ, σ�, 1 − β − β′ − ε + η − ρ, σ��.

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