A High-Order Algorithm for Time-Caputo-Tempered Partial Differential Equation with Riesz Derivatives in Two Spatial Dimensions

A novel second-order numerical approximation for the Riemann–Liouville tempered fractional derivative, called the tempered fractional-compact difference formula is derived by using the tempered Grünwald difference operator and its asymptotic expansion. Using the relationship between Riemann–Liouville and the Caputo tempered fractional derivatives, then the constructed approximation formula is applied to deal with the time-Caputo-tempered partial differential equation in time, while the spatial Riesz derivative are discretized by the fourth-order compact numerical differential formulas. By using the energy method, it is proved that the proposed algorithm to be unconditionally stable and convergent with order Oτ2+h14+h24\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}\left( \tau ^2+h_1^4+h_2^4\right) $$\end{document}, where τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is the temporal stepsize and h1,h2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_1,h_2$$\end{document} are the spatial stepsizes respectively. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm.