Anomalous diffusion for mass transport phenomena I: Analytic solutions to time fractional diffusion

TL;DR

This paper provides explicit analytic solutions for time fractional diffusion equations using special functions, generalizing classical diffusion and enabling better understanding of anomalous mass transport phenomena.

Contribution

It introduces a set of analytic solutions for time fractional diffusion problems expressed with Mittag-Leffler and M-Wright functions, extending classical solutions to anomalous diffusion.

Findings

Analytic solutions derived using Mittag-Leffler and M-Wright functions.

Time fractional diffusion generalizes classical Fickian diffusion.

Procedure to extend canonical solutions to anomalous diffusion systems.

Abstract

Mass transport problems are ubiquitous in diverse fields of physics and engineering. With the development of fractional calculus, many have taken to studying problems of fractional mass transport either through numerical simulations or through complex mathematical structures (e.g. Fox-H functions). Here, we present a set of analytic solutions to common time fractional diffusion problems, written in terms of Mittag-Leffler and M-Wright functions, as well as generalized fractional error and complementary error functions derived within. We additionally show how time fractional diffusion is a generalization of a two-parameter stretched-time fractional diffusion process. Finally we present a procedure to take canonical solutions to mass transport problems with Fickian diffusion and extend these to systems with anomalous diffusion.