Computable Solutions of Fractional Reaction-Diffusion Equations Associated with Generalized Riemann-Liouville Fractional Derivatives of Fractional Order

TL;DR

This paper derives explicit, computable solutions for fractional reaction-diffusion equations involving generalized Riemann-Liouville derivatives, extending previous work with Caputo derivatives, using Laplace and Fourier transforms and Mittag-Leffler functions.

Contribution

It introduces solutions for distributed order fractional reaction-diffusion equations with generalized Riemann-Liouville derivatives, expanding the mathematical framework beyond prior Caputo-based models.

Findings

Solutions expressed in terms of generalized Mittag-Leffler functions.

Closed-form solutions obtained via Laplace and Fourier transforms.

Extends existing literature with more general fractional derivatives.

Abstract

This paper is in continuation of the authors' recently published paper (Journal of Mathematical Physics 55(2014)083519) in which computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative is derived. In the present paper, computable solutions of distributed order fractional reaction-diffusion equations associated with generalized Riemann-Liouville derivatives of fractional orders as the time-derivative and Riesz-Feller fractional derivative as the space derivative are investigated. The solutions of the fractional reaction-diffusion equations of fractional orders are obtained in this paper. The method followed in deriving the solutions is that of joint Laplace and Fourier transforms. The solutions obtained are in a closed and computable form in terms of the familiar generalized Mittag-Leffler functions. They provide elegant extensions of the results given in the literature.