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 L₂-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is O(k^α) for a uniform time step k, where α∈(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.