Communications in Mathematical Sciences

Volume 18 (2020)

Number 7

Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

Pages: 1949 – 1975

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n7.a6

Authors

Manh Hong Duong (School of Mathematics, University of Birmingham, United Kingdom)

Bangti Jin (Department of Computer Science, University College London, United Kingdom)

Abstract

In this work, we investigate a variational formulation for a time-fractional Fokke–Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [R. Jordan, D. Kinderlehrer, and F. Otto, SIAM J. Math. Anal., 29(1):1–17, 1998]. We propose a JKO-type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.

Keywords

Wasserstein gradient flow, time-fractional Fokker–Planck equation, convergence of time-discretization scheme

2010 Mathematics Subject Classification

35Q84, 60G22, 65M12

Full Text (PDF format)

Received 1 October 2019

Accepted 10 May 2020

Published 11 December 2020