The fundamental solution of the space-time fractional diffusion equation

TL;DR

This paper investigates the fundamental solution of a space-time fractional diffusion-wave equation, analyzing its properties, probability interpretation, and explicit formulas for solution representation.

Contribution

It provides a comprehensive analysis of the fundamental solution using Mellin-Barnes integrals, extending probability interpretation to fractional derivatives.

Findings

Derived explicit formulas and asymptotic expansions

Extended probability interpretation to fractional cases

Analyzed scaling and similarity properties

Abstract

We deal with the Cauchy problem for the space-time fractional diffusion-wave equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order alpha in (0,2] and skewness theta, and the first-order time derivative with a Caputo derivative of order beta in (0,2]. The fundamental solution is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. By using the Mellin transform, we provide a general representation of the solution in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation known for the standard diffusion equation to suitable ranges of the relevant parameters alpha and beta. We derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the corresponding spatial probability densities.