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Volume 20, Issue 6
A Sharp $\alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation

Can Wang, Weihua Deng & Xiangong Tang

Int. J. Numer. Anal. Mod., 20 (2023), pp. 739-771.

Published online: 2023-11

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  • 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.

  • AMS Subject Headings

65M06, 65M12

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-739, author = {Wang , CanDeng , Weihua and Tang , Xiangong}, title = {A Sharp $\alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {6}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1033}, url = {http://global-sci.org/intro/article_detail/ijnam/22140.html} }
TY - JOUR T1 - A Sharp $\alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation AU - Wang , Can AU - Deng , Weihua AU - Tang , Xiangong JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 739 EP - 771 PY - 2023 DA - 2023/11 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1033 UR - https://global-sci.org/intro/article_detail/ijnam/22140.html KW - Fractional diffusion equation, weak singularity, middle rectangle quadrature formula, modified $L1$ scheme, five-point difference scheme, graded mesh, $α$-robust. AB -

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.

Wang , CanDeng , Weihua and Tang , Xiangong. (2023). A Sharp $\alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation. International Journal of Numerical Analysis and Modeling. 20 (6). 739-771. doi:10.4208/ijnam2023-1033
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