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

TL;DR

This paper develops and analyzes two numerical schemes for solving a time-fractional Fokker-Planck equation with space- and time-dependent forcing, demonstrating their accuracy and convergence properties through theoretical analysis and numerical experiments.

Contribution

It introduces two novel numerical schemes for the time-fractional Fokker-Planck equation and provides rigorous error analysis and validation through numerical experiments.

Findings

Space discretization is second-order accurate in $L_2$-norm.

Time-stepping scheme has an error of $O(k^eta)$ with $eta o ext{fractional order}$.

Numerical experiments confirm theoretical convergence rates.

Abstract

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 $L_2$-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is $O(k^\alpha)$ for a uniform time step $k$, where $\alpha\in(1/2,1)$ is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.