Applications of integral transforms in fractional diffusion processes

TL;DR

This paper explores the fundamental solutions of space-time fractional diffusion equations, deriving new integral representations and analyzing their properties and interpretations for advanced mathematical modeling.

Contribution

It introduces a Mellin transform-based representation of the Green function, enhancing computational and analytical understanding of fractional diffusion processes.

Findings

Derived a Mellin-Barnes integral representation of the Green function.

Analyzed the scaling and similarity properties of the fundamental solution.

Provided a probabilistic interpretation of the Green function.

Abstract

The fundamental solution (Green function) for the Cauchy problem of the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. Then, by using the Mellin transform, a general representation of the Green function in terms of Mellin-Barnes integrals in the complex plane is derived. This allows us to obtain a suitable computational form of the Green function in the space-time domain and to analyse its probability interpretation.