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Volume 42, Issue 4
Efficient Spectral Methods for Eigenvalue Problems of the Integral Fractional Laplacian on a Ball of Any Dimension

Suna Ma, Huiyuan Li, Zhimin Zhang, Hu Chen & Lizhen Chen

DOI: 10.4208/jcm.2304-m2022-0243

J. Comp. Math., 42 (2024), pp. 1032-1062.

Published online: 2024-04

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  • Abstract

An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper. The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis. And a sharp estimate on the algebraic system’s condition number is established which behaves as $N^{4s}$ with respect to the polynomial degree $N,$ where $2s$ is the fractional derivative order. The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces. Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived. Meanwhile, rigorous error estimates of the eigenvalues and eigenvectors are obtained. Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.

  • Keywords

Integral fractional Laplacian, Spectral method, Eigenvalue problem, Regularity analysis, Error estimate.

  • AMS Subject Headings

65N35, 65N25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1032, author = {Ma , SunaLi , HuiyuanZhang , ZhiminChen , Hu and Chen , Lizhen}, title = {Efficient Spectral Methods for Eigenvalue Problems of the Integral Fractional Laplacian on a Ball of Any Dimension}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {4}, pages = {1032--1062}, abstract = {

An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper. The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis. And a sharp estimate on the algebraic system’s condition number is established which behaves as $N^{4s}$ with respect to the polynomial degree $N,$ where $2s$ is the fractional derivative order. The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces. Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived. Meanwhile, rigorous error estimates of the eigenvalues and eigenvectors are obtained. Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2304-m2022-0243}, url = {http://global-sci.org/intro/article_detail/jcm/23045.html} }
TY - JOUR T1 - Efficient Spectral Methods for Eigenvalue Problems of the Integral Fractional Laplacian on a Ball of Any Dimension AU - Ma , Suna AU - Li , Huiyuan AU - Zhang , Zhimin AU - Chen , Hu AU - Chen , Lizhen JO - Journal of Computational Mathematics VL - 4 SP - 1032 EP - 1062 PY - 2024 DA - 2024/04 SN - 42 DO - http://doi.org/10.4208/jcm.2304-m2022-0243 UR - https://global-sci.org/intro/article_detail/jcm/23045.html KW - Integral fractional Laplacian, Spectral method, Eigenvalue problem, Regularity analysis, Error estimate. AB -

An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper. The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis. And a sharp estimate on the algebraic system’s condition number is established which behaves as $N^{4s}$ with respect to the polynomial degree $N,$ where $2s$ is the fractional derivative order. The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces. Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived. Meanwhile, rigorous error estimates of the eigenvalues and eigenvectors are obtained. Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.

Ma , SunaLi , HuiyuanZhang , ZhiminChen , Hu and Chen , Lizhen. (2024). Efficient Spectral Methods for Eigenvalue Problems of the Integral Fractional Laplacian on a Ball of Any Dimension. Journal of Computational Mathematics. 42 (4). 1032-1062. doi:10.4208/jcm.2304-m2022-0243
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