@Article{IJNAM-20-6,
author = {Wang, Can and Weihua, Deng and Xiangong, Tang},
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 = {<p style="text-align: justify;">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.</p>},
issn = {2617-8710},
doi = {https://doi.org/10.4208/ijnam2023-1033},
url = {https://global-sci.com/article/82915/a-sharp-alpha-robust-l1-scheme-on-graded-meshes-for-two-dimensional-time-tempered-fractional-fokker-planck-equation}
}