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

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.