Space-time fractional reaction-diffusion equations associated with a generalized Riemann-Liouville fractional derivative

TL;DR

This paper derives explicit solutions for a generalized space-time fractional reaction-diffusion equation using Laplace and Fourier transforms, extending previous models with new fractional derivatives and exploring multiple Riesz-Feller derivatives.

Contribution

It introduces a unified fractional reaction-diffusion model with generalized derivatives and provides closed-form solutions, extending prior fundamental solutions with new fractional calculus techniques.

Findings

Explicit solutions in terms of Mittag-Leffler functions

Extension of fundamental solutions for fractional diffusion

Analysis of equations with multiple Riesz-Feller derivatives

Abstract

This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann-Liouville fractional derivative defined in Hilfer et al. , and the space derivative of second order by the Riesz-Feller fractional derivative, and adding a function $\phi(x,t)$. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag-Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al., and the result very recently given by Tomovski et al.. At the end, extensions of the derived results, associated with a finite number of Riesz-Feller space fractional derivatives, are also investigated.