TY - JOUR
T1 - Finite Difference Schemes for the Tempered Fractional Laplacian
AU - Zhijiang Zhang, Weihua Deng & Hongtao Fan
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 492
EP - 516
PY - 2018
DA - 2018/12
SN - 12
DO - http://doi.org/10.4208/nmtma.OA-2017-0141
UR - https://global-sci.org/intro/article_detail/nmtma/12906.html
KW - 
AB - <p style="text-align: justify;">The second and all higher order moments of the $\beta$-stable Lévy 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&nbsp; Lévy 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., 16(1), 125-149, 2018]. 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.</p>