A time dependent fractional order diffusion equation with constant diffusivity matrix

TL;DR

This paper develops a spectral and backward Euler numerical scheme for a time-dependent fractional diffusion equation in nonhomogeneous domains, with analysis and experiments demonstrating accuracy and boundary behavior.

Contribution

It introduces a spectral approximation combined with backward Euler for fractional diffusion in nonhomogeneous domains, including error analysis and numerical validation.

Findings

The scheme accurately captures boundary behavior.

Numerical experiments confirm theoretical error estimates.

Nonhomogeneous domain effects are demonstrated.

Abstract

Of primary interest in this paper is the numerical approximation of a time dependent fractional, in space, diffusion equation where the domain is assumed to be nonhomogeneous, having different axial diffusion coefficients. This work is motivated from the consideration of composite material which can exhibit different material properties along, and perpendicular to, internal planar structures. Careful attention is paid to accurately capture the boundary behavior of the solution. A spectral approximation scheme is used for the spatial discretization and a backward Euler approximation used for the temporal discretization. Following an error analysis for the approximation scheme, numerical experiments are given to demonstrate the effects of the nonhomogeneous domain and to support the theoretical analysis.