A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing. (15th February 2019)

• Record Type: Journal Article
• Authors: Huang, Can; Le, Kim Ngan; Stynes, Martin
• Is Part Of: IMA journal of numerical analysis. Volume 40:Number 2(2020)
• Page Start: 1217
• Page End: 1240
• Publication Date: 2019-02-15
• Subjects: fractional Fokker–Planck equation -- initial-boundary value problem -- finite elements -- time-dependent forcing -- Gronwall inequality; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drz006
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal., 54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0, T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al ., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$ . Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al .'s paper for the semidiscrete (backward Euler scheme in time) method.