Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension

TL;DR

This paper develops a spectral method using Jacobi polynomials to approximate solutions of variable coefficient fractional diffusion equations in one dimension, providing error analysis and numerical validation.

Contribution

Introduces a spectral approximation scheme for variable coefficient fractional diffusion equations by transforming them into constant coefficient problems and analyzing the error.

Findings

Error estimates are sharp and validated numerically.

Spectral method effectively approximates the solution to variable coefficient FDEs.

Numerical experiments confirm theoretical error bounds.

Abstract

In this article we consider the approximation of a variable coefficient (two-sided) fractional diffusion equation (FDE), having unknown $u$. By introducing an intermediate unknown, $q$, the variable coefficient FDE is rewritten as a lower order, constant coefficient FDE. A spectral approximation scheme, using Jacobi polynomials, is presented for the approximation of $q$, $q_{N}$. The approximate solution to $u$, $u_{N}$, is obtained by post processing $q_{N}$. An a priori error analysis is given for $(q \, - \, q_{N})$ and $(u \, - \, u_{N})$. Two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates.