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

Marvin Fritz

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arXiv · 2307.16615

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

Marvin Fritz

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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.

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Marvin Fritz. 2023-07-31. Well-posedness and simulation of weak solutions to the time-fractional Fokker-Planck equation with general forcing. https://arxiv.org/abs/2307.16615

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