Finite difference/spectral approximation for a time–space fractional equation on two and three space dimensions

We consider the numerical method for solving a time–space fractional equation whose solution has a weak initial singularity. L1 scheme on graded mesh is used to capture the initial weak singularity. It is shown that the method can recover order 2−α convergence by choosing some appropriate grading parameters, where α (0<α<1) is the order of temporal Caputo fractional derivative. A fully discrete scheme is proposed by combining spectral approximation for the discretization of space in 2D and 3D cases. Stability and convergence of the fully discrete scheme are rigorously established. Numerical results are presented to show the sharpness of the error estimate.