H. J. Haubold1,2*, A. M. Mathai2,3 and R. K.
Saxena4
1Office for Outer Space Affairs, United Nations, Vienna International
Centre, P.O. Box 500,
A-1400, Vienna, Austria
2Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O., Pala
686 574, Kerala,
India
3Department of Mathematics and Statistics, McGil l University,
Montreal, Canada H3A 2K6
4Department of Mathematics and Statistics, Jai Narain Vyas University,
Jodhpur 342 004,
India
Abstract. This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space- time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction-diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation contain- ing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space- fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form.
Keywords: fractional calculus - reaction-diffusion equations - Fox's H-function - Caputo derivative - Riesz-Feller derivative