Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function
We discuss and derive the analytical solution for the fractional partial differential equation with generalized Riemann-Liouville fractional operator of order and . Here, we derive the solution of the given differential equation with the help of Laplace and Hankel transform in terms of Fox's -function as well as in terms of Fox-Wright function .
1. 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 and the left-sided Riemann-Liouville fractional integral operator , and the corresponding Riemann-Liouville fractional derivative operators and , as follows [10, 11]:
where the function is locally integrable, denotes the real part of the complex number and means the greatest integer in .
Definition 1.1. The right-sided fractional derivative and the left-sided fractional derivative of order and type with respect to are defined by
whenever the second number of (1.4) exists. This generalization (1.4) yields the classical Riemann-Liouville fractional derivative operator when . Moreover, for , it gives the fractional derivative operator introduced by Liouville  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 are given .
Using the formulas (1.1) and (1.2) in conjunction with (1.3) when , the fractional derivative operator can be written in the following form: The difference between fractional derivatives of different types becomes apparent from their Laplace transformations. For example, it is found for that [1, 2, 9] where is the Riemann-Liouville fractional integral of order evaluated in the limit as , it being understood (as usual) that , provided that the defining integral in (1.7) exists.
The familiar Mittag-Leffler functions and are defined by the following series: respectively. These functions are natural extensions of the exponential, hyperbolic, and trigonometric functions, since
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.  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 of (1.2) was introduced by Prabhakar  as follows: where denotes the familiar Pochhammer symbol, defined (for and in terms of the familiar Gamma function) by
Clearly, we have the following special cases: Indeed, as already observed earlier by Srivastava and Saxena , the generalized Mittag-Leffler function itself is actually a very specialized case of a rather extensively investigated function as indicated below : Here and in what follows, denotes the Wright (or more appropriately, the Fox-Wright) generalized of the hypergeometric function, which is defined as follows : in which we assumed in general that
In application of Mittag-Leffler function, it is useful to have the following Laplace inverse transform formula: where .
2. Fox’s -function
The Fox function, also referred as the Fox’s -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 -function is defined as 
We need this relation
3. Finite Hankel Transform
If satisfies Dirichlet conditions in closed interval and if its finite Hankel transform is defined to be 
where are the roots of the equation . Then at each point of the interval at which is continuous:
where the sum is taken over all positive roots of , and are Bessel functions of first kind.
In application of the finite Hankel transform to physical problems, it is useful to have the following formula 
Example 3.1. Solve the differential equation
with initial condition
Solution 1. Taking Laplace transform of (3.4), we get
Taking Hankel transform on both side of the above equation, we get
then we get
On taking Laplace inverse of (3.10), (3.11), and (3.12), respectively, After taking Inverse Laplace and Hankel transform of (3.9) put the value (3.13) through (3.15) in (3.9), we get
Example 3.2. Solve the differential equation (3.4) with initial condition
Solution 2. Taking Laplace and Hankel transform of (3.4), we get
on taking Inverse Laplace transform of equation (3.19), we get
By using convolution theorem for Laplace transform and taking inverse Hankel transform, we get or By using the relation (2.2) or which is the required solution.
R. Hilfer, “Threefold Introduction to fractional derivatives,” in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons, and I. M. Sokolov, Eds., pp. 17–73, Wiley-VCH Verlag, Weinheim, Germany, 2008.View at: Google Scholar
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, vol. 2, McGraw-Hill, London, UK, 1954.View at: Zentralblatt MATH
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York, NY, USA, 1993.
J. Liouville, “Mémoire sur quelques quéstions de géometrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces quéstions,” École Polytechnique, vol. 13, no. 21, pp. 1–69, 1832.View at: Google Scholar
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, vol. 1, McGraw-Hill, London, UK, 1954.View at: Zentralblatt MATH
R. Gorenflo, F. Mainardi, and H. M. Srivastava, “Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena,” in Proceedings of the 8th International Colloquium on Differential Equations, D. Bainov, Ed., pp. 195–202, VSP, Plovdiv, Bulgaria, August 1997.View at: Google Scholar | Zentralblatt MATH
A. A. Kilbas, M. Saigo, and R. K. Saxena, “Solution of Volterra integrodifferential equations with generalized Mittag-Leffler function in the kernels,” Journal of Integral Equations and Applications, vol. 14, no. 4, pp. 377–396, 2002.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
H. M. Srivastava and R. K. Saxena, “Some Volterra-type fractional integro-differential equations with a multivariable confluent hypergeometric function as their kernel,” Journal of Integral Equations and Applications, vol. 17, no. 2, pp. 199–217, 2005.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, NY, USA, 1951.