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Volume 12, Issue 2
Finite Difference Schemes for the Tempered Fractional Laplacian

Zhijiang Zhang, Weihua Deng & Hongtao Fan

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 492-516.

Published online: 2018-12

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  • Abstract

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  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.

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@Article{NMTMA-12-492, author = {}, title = {Finite Difference Schemes for the Tempered Fractional Laplacian}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {2}, pages = {492--516}, abstract = {

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  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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2017-0141}, url = {http://global-sci.org/intro/article_detail/nmtma/12906.html} }
TY - JOUR T1 - Finite Difference Schemes for the Tempered Fractional Laplacian 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 -

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  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.

Zhijiang Zhang, Weihua Deng & Hongtao Fan. (2020). Finite Difference Schemes for the Tempered Fractional Laplacian. Numerical Mathematics: Theory, Methods and Applications. 12 (2). 492-516. doi:10.4208/nmtma.OA-2017-0141
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