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

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.

3 Figures

References in corpus (3)

• Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces
• Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving
• Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

Cited by in corpus (9)

• On energy dissipation theory and numerical stability for time-fractional phase field equations
• Regularity theory for time-fractional advection-diffusion-reaction equations
• Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing
• A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data
• Analysis of a dilute polymer model with a time-fractional derivative
• Wasserstein Gradient Flow Formulation of the Time-Fractional Fokker-Planck Equation
• A finite difference method for space fractional differential equations with variable diffusivity coefficient
• Optimal order finite element approximations for variable-order time-fractional diffusion equations
• Uniform stability for a spatially-discrete, subdiffusive Fokker-Planck equation