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

Jin, B; Duong, MH; (2020) Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation. Communications in Mathematical Sciences , 18 (7) pp. 1949-1975. 10.4310/CMS.2020.v18.n7.a6. Green open access

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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.

Type: Article
Title: Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation
Open access status: An open access version is available from UCL Discovery
DOI: 10.4310/CMS.2020.v18.n7.a6
Publisher version: https://doi.org/10.4310/CMS.2020.v18.n7.a6
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Wasserstein gradient flow, time-fractional Fokker–Planck equation, convergence of time-discretization scheme
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10097078
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