An operator splitting scheme for the fractional kinetic Fokker-Planck equation

Manh Hong Duong; Yulong Lu

doi:10.3934/dcds.2019250

Article Contents

2019, Volume 39, Issue 10: 5707-5727. Doi: 10.3934/dcds.2019250

This issue Previous Article Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching Next Article On the maximal saddle order of

$ p:-q $

resonant saddle

An operator splitting scheme for the fractional kinetic Fokker-Planck equation

Manh Hong Duong^1, and
Yulong Lu^2, ,

1.
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
2.
Department of Mathematics, Duke University, Durham NC 27708, USA

^* Corresponding author: Yulong Lu
^* Corresponding author: Yulong Lu

Received: June 2018

Revised: March 2019

Published: July 2019

MHD was supported by ERC Starting Grant 335120

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Abstract

In this paper, we develop an operator splitting scheme for the fractional kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a fractional diffusion phase and a kinetic transport phase. The first phase is solved exactly using the convolution operator while the second one is solved approximately using a variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. We prove the convergence of the scheme to a weak solution to FKFPE. As a by-product of our analysis, we also establish a variational formulation for a kinetic transport equation that is relevant in the second phase. Finally, we discuss some extensions of our analysis to more complex systems.

Keywords:

Operator splitting,
variational method,
fractional kinetic Fokker-Planck equation,
kinetic transport equation,
optimal transportation.

Mathematics Subject Classification: Primary: 49S05, 35Q84; Secondary: 49J40.

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