Computational solutions of distributed oder reaction-diffusion systems associated with Riemann-Liouville derivatives

TL;DR

This paper develops computational solutions for distributed order fractional reaction-diffusion equations involving Riemann-Liouville derivatives, extending previous work by deriving closed-form solutions using Laplace and Fourier transforms with Mittag-Leffler functions.

Contribution

It introduces a novel method for solving distributed order fractional reaction-diffusion equations with Riemann-Liouville derivatives, providing explicit solutions in terms of hypergeometric series.

Findings

Closed-form solutions in terms of Mittag-Leffler functions

Extension of previous models to Riemann-Liouville derivatives

Convergence analysis of the series solutions

Abstract

This article is in continuation of our earlier article [37] in which computational solution 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 this article, we present computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of the results given earlier by Chen et al. [1], Debnath [3], Saxena et al. [36], Haubold et al. [15] and Pagnini and Mainardi [30]. The results obtained are presented in the form of two theorems. Some interesting results associated with fractional Riesz derivatives are also derived as special cases of our findings. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kamp\'e de F\'eriet hypergeometric series in two variables, defined by Srivastava and Daoust [46] (also see Appendix B). The convergence of the double series occurring in the solution is also given.