Numerical algorithms for the time-space tempered fractional Fokker-Planck equation

This paper aims to provide the high order numerical schemes for the time-space tempered fractional Fokker-Planck equation in a finite domain. The high order difference operators, called the tempered and weighted and shifted Lubich difference operators, are used to approximate the time tempered fractional derivative. The spatial operators are discretized by the central difference methods. We apply the central difference methods to the spatial operators and obtain that the numerical schemes are convergent with orders O(τq+h2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\tau^{q} + h^{2})$\end{document} (q=1,2,3,4,5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q = 1,2,3,4,5$\end{document}). The stability and convergence of the first order numerical scheme are rigorously analyzed. And the effectiveness of the presented schemes is testified with several numerical experiments. Additionally, some physical properties of this diffusion system are simulated.