Abstract: This paper provides a fnite-diference discretization for the one- and two-dimensional tempered fractional Laplacian and solves the tempered fractional Poisson equation with homogeneous Dirichlet boundary conditions. The main ideas are to, respectively, use linear and quadratic interpolations to approximate the singularity and non-singularity of the one-dimensional tempered fractional Laplacian and bilinear and biquadratic interpolations to the two-dimensional tempered fractional Laplacian. Then, we give the truncation errors and prove the convergence. Numerical experiments verify the convergence rates of the order O(h^2-2s).

Key words: Tempered fractional Laplacian, Finite-diference scheme, Linear and quadratic interpolations, Bilinear and biquadratic interpolations, Convergence rates