Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian

TL;DR

This paper develops finite difference methods for the two-dimensional tempered fractional Laplacian, enabling accurate numerical solutions to related Poisson equations with verified convergence and effectiveness.

Contribution

It introduces a finite difference discretization for the 2D tempered fractional Laplacian and analyzes its error estimates for solving Poisson equations.

Findings

Numerical schemes achieve expected convergence rates.

Methods effectively solve tempered fractional Poisson problems.

Error estimates confirm scheme accuracy.

Abstract

Tempered fractional Laplacian is the generator of the tempered isotropic L\'evy process [W.H. Deng, B.Y. Li, W.Y. Tian, and P.W. Zhang, Multiscale Model. Simul., 16(1), 125-149, 2018]. This paper provides the finite difference discretization for the two dimensional tempered fractional Laplacian $(\Delta+\lambda)^{\frac{\beta}{2}}$. Then we use it to solve the tempered fractional Poisson equation with Dirichlet boundary conditions and derive the error estimates. Numerical experiments verify the convergence rates and effectiveness of the schemes.