Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function

Applications of fractional calculus require fractional derivatives of different kinds 1– 9 . Differentiation and integration of fractional order are traditionally defined by the right-sided Riemann-Liouville fractional integral operator I a and the left-sided RiemannLiouville fractional integral operator I a−, and the corresponding Riemann-Liouville fractional derivative operators D a and D P a−, as follows 10, 11 :


Introduction, Definition, and Preliminaries
Applications of fractional calculus require fractional derivatives of different kinds 1-9 .Differentiation and integration of fractional order are traditionally defined by the right-sided Riemann-Liouville fractional integral operator I P a and the left-sided Riemann-Liouville fractional integral operator I P a− , and the corresponding Riemann-Liouville fractional derivative operators D P a and D P a− , as follows 10, 11 : where the function f is locally integrable, R μ denotes the real part of the complex number μ ∈ C and R μ means the greatest integer in R μ .
Recently, a remarkable large family of generalized Riemann-Liouville fractional derivatives of order α 0 < α < 1 and type β 0 ≤ β ≤ 1 was introduced as follows 1-3, 5, 6, 8 .The difference between fractional derivatives of different types becomes apparent from their Laplace transformations.For example, it is found for 0 < α < 1 that 1, 2, 9 where f 0 is the Riemann-Liouville fractional integral of order 1 − β 1 − α evaluated in the limit as t → 0 , it being understood as usual that 13 , provided that the defining integral in 1.7 exists.
The familiar Mittag-Leffler functions E μ z and E μ,ν z are defined by the following series: respectively.These functions are natural extensions of the exponential, hyperbolic, and trigonometric functions, since

1.10
For a detailed account of the various properties, generalizations, and applications of the Mittag-Leffler functions, the reader may refer to the recent works by, for example, Gorenflo et al. 14 where λ ν denotes the familiar Pochhammer symbol, defined for λ, ν ∈ C and in terms of the familiar Gamma function by

1.12
Clearly, we have the following special cases: Indeed, as already observed earlier by Srivastava and Saxena 21 , the generalized Mittag-Leffler function E λ μ,ν z itself is actually a very specialized case of a rather extensively investigated function p Ψ q as indicated below 17 : Here and in what follows, p Ψ q denotes the Wright or more appropriately, the Fox-Wright generalized of the hypergeometric p F q function, which is defined as follows 12 : in which we assumed in general that a j , A j ∈ C j 1, . . ., p , b j , B j ∈ C j 1, . . ., q .1.17 In application of Mittag-Leffler function, it is useful to have the following Laplace inverse transform formula: where E j γ,β z d j /dz j E γ,β z .

Fox's H-function
The Fox function, also referred as the Fox's H-function, generalizes the Mellin-Barnes function.The importance of the Fox function lies in the fact that it includes nearly all special functions occurring in applied mathematics and statistics as special cases.Fox H-function is defined as 22 We need this relation 2.2

Finite Hankel Transform
If f r satisfies Dirichlet conditions in closed interval 0, a and if its finite Hankel transform is defined to be 23 where λ n are the roots of the equation J 0 r 0. Then at each point of the interval at which f r is continuous: where the sum is taken over all positive roots of J 0 r 0, J 0 and J 1 are Bessel functions of first kind.
In application of the finite Hankel transform to physical problems, it is useful to have the following formula 23 where 0 < α ≤ 1/2 and 0 ≤ β ≤ 1 with initial condition u r, t 0 everywhere for t < 0, u r, t 0 for r 1, t > 0, u r, t finite at r 0, t > 0.

3.5
Solution 1. Taking Laplace transform of 3.4 , we get 1 r u r, s f s .

3.6
Taking Hankel transform on both side of the above equation, we get where 3.11

3.15
After taking Inverse Laplace and Hankel transform of 3.9 put the value 3.13 through 3.15 in 3.9 , we get 3.17

Definition 1 . 1 .
The right-sided fractional derivative D α,β a and the left-sided fractional derivative D α,β a− of order α 0 < α < 1 and type β 0 ≤ β ≤ 1 with respect to x are defined by number of 1.4 exists.This generalization 1.4 yields the classical Riemann-Liouville fractional derivative operator when β 0.Moreover, for β 1, it gives the fractional derivative operator introduced by Liouville 12 which is often attributed to Caputo now-a-days and which should more appropriately be referred to as the Liouville-Caputo fractional derivative.Several authors 7, 9 called the general operators in 1.4 the Hilfer fractional derivative operators.Applications of D α,β a± are given 3 .Using the formulas 1.1 and 1.2 in conjunction with 1.3 when n 1, the fractional derivative operator D α,β a± can be written in the following form:
and Kilbas et al. 15-17 .The Mittag-Leffler function 1.1 and some of its various generalizations have only recently been calculated numerically in the whole complex plane 18, 19 .By means of the series representation, a generalization of the Mittag-Leffler function E μ,ν z of 1.2 was introduced by Prabhakar 20 as follows: