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. 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. The discrete fractional Gronwall 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.