Convergence Analysis of a LDG Method for Time–Space Tempered Fractional Diffusion Equations with Weakly Singular Solutions

A class of time–space tempered fractional diffusion equations is considered in this paper. The solution of these problems generally have a weak singularity near the initial time t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 0$$\end{document}. To solve the time–space tempered fractional diffusion equations, a fully discrete local discontinuous Galerkin (LDG) method is proposed. The basic idea is to apply LDG method in the space on uniform meshes and a finite difference method in the time on graded meshes to deal with the weak singularity at initial time t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 0$$\end{document}. The discrete fractional Grönwall inequality is used to analyze the stability and convergence of the method. Numerical results show that the proposed method for time–space tempered fractional diffusion equation is accurate and reliable.