Computable solutions of fractional partial differential equations related to reaction-diffusion systems

TL;DR

This paper derives a computable, closed-form solution for fractional partial differential equations involving Riemann-Liouville and Riesz-Feller derivatives, extending previous results using Laplace and Fourier transforms and H-functions.

Contribution

It presents a novel method to obtain explicit solutions for fractional PDEs related to reaction-diffusion systems, including special cases like fractional Schrödinger and diffusion-wave equations.

Findings

Solution expressed in terms of H-function

Derived four theorems providing the solution framework

Extended previous results to new fractional PDEs

Abstract

The object of this paper is to present a computable solution of a fractional partial differential equation associated with a Riemann-Liouville derivative of fractional order as the time-derivative and Riesz-Feller fractional derivative as the space derivative. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computable form in terms of the H-function. It provides an elegant extension of the results given earlier by Debnath, Chen et al., Haubold et al., Mainardi et al., Saxena et al., and Pagnini et al. The results obtained are presented in the form of four theorems. Some results associated with fractional Schroeodinger equation and fractional diffusion-wave equation are also derived as special cases of the findings.