The role of the Fox-Wright functions in fractional sub-diffusion of distributed order

TL;DR

This paper derives the fundamental solution of distributed order fractional diffusion equations using Mellin-Barnes integrals, linking it to Fox-Wright functions, and explores the distribution of time-scales in sub-diffusion processes.

Contribution

It introduces a novel integral representation of solutions for distributed order fractional diffusion equations using Fox-Wright functions.

Findings

Solution expressed via Mellin-Barnes integral

Series expansion reveals time-scale distribution

Re-derivation of single-order fractional diffusion results

Abstract

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.