Numerical solution of the time-fractional Fokker-Planck equation with general forcing (Q2814456)

This paper deals with two schemes to solve a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The discretization in these schemes is either in space using a piecewise-linear Galerkin finite element method or in time using a time-stepping procedure similar to the classical implicit Euler method. The scheme arising from the discretization in space is proved to be of order two in \(L^\infty(L^2)\)-norm. For the time-stepping scheme, it is shown that the order is \(O(k^\alpha)\) for a uniform time step \(k\), where \(\alpha\in(1/2,1)\) is the fractional diffusion parameter. Some numerical tests are presented for the fully discrete scheme which combines the above stated discretizations and it is found that convergence behavior confirms the theoretical results.