NUMERICAL SOLUTION OF THE TIME-FRACTIONAL FOKKER-PLANCK EQUATION WITH GENERAL FORCING

We study two schemes for a time-fractional Fokker-Planck equation with space-and time-dependent forcing in one space dimension. The first scheme is continuous in time and discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial L-2-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is O(k(alpha)) for a uniform time step k, where alpha epsilon (1/2, 1) is the fractional diffusion parameter. In numerical experiments using a combined, fully discrete method, we observe convergence behavior consistent with these results.