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

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, qN. The approximate solution to u, uN, is obtained by post processing qN. An a priori error analysis is given for (q−qN) and (u−uN). Two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates.