Well-posedness and simulation of weak solutions to the time-fractional Fokker-Planck equation with general forcing

TL;DR

This paper establishes the well-posedness of weak solutions to the time-fractional Fokker-Planck equation with general forces, introducing novel analytical techniques and a numerical scheme for simulations of anomalous diffusion.

Contribution

It provides the first comprehensive analysis of well-posedness for this equation with space-time dependent forces and develops a new numerical algorithm for simulations.

Findings

Energy inequality derived using novel testing methods

Proposed nonuniform L1 scheme for numerical approximation

Simulation results demonstrate effectiveness for various forces

Abstract

In this paper, we investigate the well-posedness of weak solutions to the time-fractional Fokker-Planck equation. Its dynamics is governed by anomalous diffusion, and we consider the most general case of space-time dependent forces. Consequently, the fractional derivatives appear on the right-hand side of the equation, and they cannot be brought to the left-hand side, which would have been preferable from an analytical perspective. For showing the model's well-posedness, we derive an energy inequality by considering nonstandard and novel testing methods that involve a series of convolutions and integrations. We close the estimate by a Henry-Gronwall-type inequality. Lastly, we propose a numerical algorithm based on a nonuniform L1 scheme and present some simulation results for various forces.