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 is 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, whereas the corresponding error for the time-stepping scheme is of order 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 behaviour consistent with these results.