# Solutions of fractional reaction-diffusion equations in terms of the H-function

## TL;DR

This paper derives a general solution for fractional reaction-diffusion equations using H-functions, encompassing various special cases like space-time and neutral fractional diffusion, through Laplace and Fourier transforms.

## Contribution

It introduces a unified approach to solving fractional reaction-diffusion equations with Riesz-Feller derivatives, generalizing previous results and providing solutions in terms of H-functions.

## Key findings

- Solution expressed in closed form using H-functions
- Includes fundamental solutions for various fractional diffusion types
- Generalizes earlier results by Mainardi and Saxena

## Abstract

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction- diffusion equations. The advantage of using 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 neutral fractional diffusion, space-fractional 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-functions in compact form.