Finite difference schemes for the tempered fractional Laplacian

TL;DR

This paper develops finite difference schemes for solving equations involving the tempered fractional Laplacian, addressing accuracy and computational techniques, with validation through numerical examples.

Contribution

It introduces novel finite difference schemes for the tempered fractional Laplacian and discusses their accuracy and efficient solution methods.

Findings

Schemes achieve high accuracy depending on solution regularity

Effective algorithms for solving algebraic systems are presented

Numerical examples demonstrate the schemes' performance

Abstract

The second and all higher order moments of the $\beta$-stable L\'{e}vy process diverge, the feature of which is sometimes referred to as shortcoming of the model when applied to physical processes. So, a parameter $\lambda$ is introduced to exponentially temper the L\'{e}vy process. The generator of the new process is tempered fractional Laplacian $(\Delta+\lambda)^{\beta/2}$ [W.H. Deng, B.Y. Li, W.Y. Tian, and P.W. Zhang, Multiscale Model. Simul., in press, 2017]. In this paper, we first design the finite difference schemes for the tempered fractional Laplacian equation with the generalized Dirichlet type boundary condition, their accuracy depending on the regularity of the exact solution on $\bar{\Omega}$. Then the techniques of effectively solving the resulting algebraic equation are presented, and the performances of the schemes are demonstrated by several numerical examples.