Volume 13, Issue 1
Error Estimates of Finite Difference Methods for the Fractional Poisson Equation with Extended Nonhomogeneous Boundary Conditions

East Asian J. Appl. Math., 13 (2023), pp. 194-212.

Published online: 2023-01

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• Abstract

Two efficient finite difference methods for the fractional Poisson equation involving the integral fractional Laplacian with extended nonhomogeneous boundary conditions are developed and analyzed. The first one uses appropriate numerical quadratures to handle extended nonhomogeneous boundary conditions and weighted trapezoidal rule with a splitting parameter to approximate the hypersingular integral in the fractional Laplacian. It is proven that the method converges with the second-order accuracy provided that the exact solution is sufficiently smooth and a splitting parameter is suitably chosen. Secondly, if numerical quadratures fail, we employ a truncated based method. Under specific conditions, the convergence rate of this method is optimal, as error estimates show. Numerical experiments are provided to gauge the performance of the methods proposed.

35R11, 65M06

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@Article{EAJAM-13-194, author = {Li , Xinyan}, title = {Error Estimates of Finite Difference Methods for the Fractional Poisson Equation with Extended Nonhomogeneous Boundary Conditions}, journal = {East Asian Journal on Applied Mathematics}, year = {2023}, volume = {13}, number = {1}, pages = {194--212}, abstract = {

Two efficient finite difference methods for the fractional Poisson equation involving the integral fractional Laplacian with extended nonhomogeneous boundary conditions are developed and analyzed. The first one uses appropriate numerical quadratures to handle extended nonhomogeneous boundary conditions and weighted trapezoidal rule with a splitting parameter to approximate the hypersingular integral in the fractional Laplacian. It is proven that the method converges with the second-order accuracy provided that the exact solution is sufficiently smooth and a splitting parameter is suitably chosen. Secondly, if numerical quadratures fail, we employ a truncated based method. Under specific conditions, the convergence rate of this method is optimal, as error estimates show. Numerical experiments are provided to gauge the performance of the methods proposed.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.070422.220922}, url = {http://global-sci.org/intro/article_detail/eajam/21309.html} }
TY - JOUR T1 - Error Estimates of Finite Difference Methods for the Fractional Poisson Equation with Extended Nonhomogeneous Boundary Conditions AU - Li , Xinyan JO - East Asian Journal on Applied Mathematics VL - 1 SP - 194 EP - 212 PY - 2023 DA - 2023/01 SN - 13 DO - http://doi.org/10.4208/eajam.070422.220922 UR - https://global-sci.org/intro/article_detail/eajam/21309.html KW - Fractional Poisson equation, finite difference method, nonhomogeneous boundary condition, error estimates, integral fractional Laplacian. AB -

Two efficient finite difference methods for the fractional Poisson equation involving the integral fractional Laplacian with extended nonhomogeneous boundary conditions are developed and analyzed. The first one uses appropriate numerical quadratures to handle extended nonhomogeneous boundary conditions and weighted trapezoidal rule with a splitting parameter to approximate the hypersingular integral in the fractional Laplacian. It is proven that the method converges with the second-order accuracy provided that the exact solution is sufficiently smooth and a splitting parameter is suitably chosen. Secondly, if numerical quadratures fail, we employ a truncated based method. Under specific conditions, the convergence rate of this method is optimal, as error estimates show. Numerical experiments are provided to gauge the performance of the methods proposed.

Xinyan Li. (2023). Error Estimates of Finite Difference Methods for the Fractional Poisson Equation with Extended Nonhomogeneous Boundary Conditions. East Asian Journal on Applied Mathematics. 13 (1). 194-212. doi:10.4208/eajam.070422.220922
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