Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution

TL;DR

This paper derives explicit solutions to fractional diffusion equations using Mellin convolutions of generalized Gamma densities, connecting special functions and stochastic processes.

Contribution

It introduces a novel approach to solving fractional diffusion equations through Mellin convolution techniques involving generalized Gamma densities.

Findings

Explicit solutions expressed via modified Bessel functions

Connection established between fractional diffusion and Bessel processes

Method applicable to a class of time-fractional PDEs

Abstract

In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process.