Anisotropic solutions of the time-fractional diffusion equation in multiple dimensions

TL;DR

This paper explores anisotropic time-fractional diffusion equations in multiple dimensions, providing solutions expressed via Wright functions and demonstrating how multidimensional cases relate to one-dimensional solutions.

Contribution

It introduces a framework for solving anisotropic fractional diffusion equations in multiple dimensions using Wright functions and transformations from the one-dimensional case.

Findings

Solutions expressed in terms of Wright functions and derivatives.

Multidimensional solutions can be derived from one-dimensional cases.

Applicable to modeling anomalous diffusion in complex media.

Abstract

Anomalous diffusion phenomena are ubiquitous in complex media, such as biological tissues. A wide class of sub-diffusive phenomena phenomena is described by the time-fractional diffusion equation. The paper investigates the case of anisotropic fractional diffusion in the Euclidean space. The solution of the fractional sub-diffusion equation can be expressed in terms of the Wright function and its spatial derivatives, parametrized by the directional unit vector (or alternatively a normal hyperplane). Moreover, the multidimensional case could be expressed as a transformation of the one-dimensional case.