A Sharp $\alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation
Year: 2023
Author: Can Wang, Weihua Deng, Xiangong Tang
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 6 : pp. 739–771
Abstract
In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with the tempered fractional derivative of order $α.$ Although some of its variants are considered in many recent numerical analysis works, there are still some significant differences. Here we first provide the regularity estimates of the solution. Then a modified $L1$ scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at $t → 0^+,$ while the five-point difference scheme is used in space. Stability and convergence are proved in the sense of $L^∞$ norm, getting a sharp error estimate $\mathscr{O}(\tau^{{\rm min}\{2−α,rα\}})$ on graded meshes. Furthermore, the constant multipliers in the analysis do not blow up as the order of Caputo fractional derivative $α$ approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence orders of the presented schemes.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2023-1033
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 6 : pp. 739–771
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 33
Keywords: Fractional diffusion equation weak singularity middle rectangle quadrature formula modified $L1$ scheme five-point difference scheme graded mesh $α$-robust.
Author Details
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A space-time Galerkin Müntz spectral method for the time fractional Fokker–Planck equation
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https://doi.org/10.1080/00207160.2024.2332957 [Citations: 0]