Distributed order reaction-diffusion systems associated with Caputo derivatives

TL;DR

This paper derives a general solution for a distributed order fractional reaction-diffusion equation involving Caputo derivatives and Riesz-Feller derivatives, unifying and extending previous models with solutions expressed via H-functions.

Contribution

It introduces a unified framework for solving distributed order fractional reaction-diffusion equations with Caputo and Riesz-Feller derivatives, including convergence analysis and special function representations.

Findings

Solution expressed in terms of H-functions

Includes fundamental solutions for generalized fractional diffusion

Convergence conditions for series solutions analyzed

Abstract

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. [23,24], for the fundamental solution of the space-time fractional equation, including Haubold et al. [13] and Saxena et al. [38] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction 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-function in compact forms. The convergence conditions for the double series occurring in the solutions are investigated. It is interesting to observe that the double series comes out to be a special case of the Srivastava-Daoust hypergeometric function of two variables given in the Appendix B of this paper.