Research Article | Open Access

V. B. L. Chaurasia, Ravi Shanker Dubey, "Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function", *International Scholarly Research Notices*, vol. 2011, Article ID 682381, 9 pages, 2011. https://doi.org/10.5402/2011/682381

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

**Academic Editor:**M. F. El-Sayed

#### Abstract

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 .

Recently, a remarkable large family of generalized Riemann-Liouville fractional derivatives of order and type was introduced as follows [1–3, 5, 6, 8].

*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 [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 are given [3].

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 [13],
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. [14] 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 [20] 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 [21], the generalized Mittag-Leffler function itself is actually a very specialized case of a rather extensively investigated function as indicated below [17]:
Here and in what follows, denotes the Wright (or more appropriately, the Fox-Wright) generalized of the hypergeometric function, which is defined as follows [12]:
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 [22]

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 [23]

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 [23]

*Example 3.1. *Solve the differential equation
where and

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
where

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.

#### References

- R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific Publishing, River Edge, NJ, USA, 2000. View at: Publisher Site | Zentralblatt MATH - R. Hilfer, “Fractional time evolution,” in
*Applications of Fractional Calculus in Physics*, R. Hilfer, Ed., pp. 87–130, World Scientific Publishing, River Edge, NJ, USA, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Hilfer, “Experimental evidence for fractional time evolution in glass forming materials,”
*Chemical Physics*, vol. 284, no. 1-2, pp. 399–408, 2002. View at: Publisher Site | Google Scholar - 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 - R. Hilfer and L. Anton, “Fractional master equations and fractal time random walks,”
*Physical Review E*, vol. 51, no. 2, pp. R848–R851, 1995. View at: Publisher Site | Google Scholar - R. Hilfer, Y. Luchko, and Ž. Tomovski, “Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives,”
*Fractional Calculus & Applied Analysis*, vol. 12, no. 3, pp. 299–318, 2009. View at: Google Scholar | Zentralblatt MATH - F. Mainardi and R. Gorenflo, “Time-fractional derivatives in relaxation processes: a tutorial survey,”
*Fractional Calculus and Applied Analysis*, vol. 10, no. 3, pp. 269–308, 2007. View at: Google Scholar | Zentralblatt MATH - T. Sandev and Ž. Tomovski, “The general time fractional wave equation for a vibrating string,”
*Journal of Physics. A. Mathematical and Theoretical*, vol. 43, no. 5, 2010. View at: Publisher Site | Google Scholar - H. M. Srivastava and Ž. Tomovski, “Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel,”
*Applied Mathematics and Computation*, vol. 211, no. 1, pp. 198–210, 2009. View at: Publisher Site | 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, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,”
*Integral Transforms and Special Functions*, vol. 15, no. 1, pp. 31–49, 2004. 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. - R. Hilfer and H. Seybold, “Computation of the generalized Mittag-Leffler function and its inverse in the complex plane,”
*Integral Transforms and Special Functions*, vol. 17, no. 9, pp. 637–652, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. J. Seybold and R. Hilfer, “Numerical results for the generalized Mittag-Leffler function,”
*Fractional Calculus and Applied Analysis*, vol. 8, no. 2, pp. 127–139, 2005. View at: Google Scholar | Zentralblatt MATH - T. R. Prabhakar, “A singular integral equation with a generalized Mittag Leffler function in the kernel,”
*Yokohama Mathematical Journal*, vol. 19, pp. 7–15, 1971. View at: Google Scholar | Zentralblatt MATH - 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 - R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,”
*Physics Reports*, vol. 339, no. 1, p. 77, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. N. Sneddon,
*Fourier Transforms*, McGraw-Hill, New York, NY, USA, 1951.

#### Copyright

Copyright © 2011 V. B. L. Chaurasia and Ravi Shanker Dubey. This 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.