Spectral and pseudospectral approximations for the time fractional diffusion equation on an unbounded domain

In this paper, we consider the numerical approximation of the time fractional diffusion equation with variable coefficients on a semi-infinite spatial domain. A fully discrete scheme based on finite difference method in time and spectral approximation using Laguerre functions in space is proposed. Stability and convergence of the proposed scheme are rigorously established. The scheme is unconditionally stable and convergent with order O(τ2+N(1−m)/2), where ττ, NN, and mm are the time-step size, polynomial degree, and regularity in the space variable of the exact solution, respectively. A pseudospectral scheme is also proposed and analysed. Some numerical examples are presented to demonstrate the theoretical results.