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Volume 15, Issue 4
A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels

Chao Zhang, Guoqing Yao & Sheng Chen

DOI: 10.4208/nmtma.OA-2022-0006s

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 1041-1062.

Published online: 2022-10

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  • Abstract

In this paper, we propose a hybrid spectral method for a type of nonlocal problems, nonlinear Volterra integral equations (VIEs) of the second kind. The main idea is to use the shifted generalized Log orthogonal functions (GLOFs) as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals. This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions. The well-posedness and the related error estimates will be provided. Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.

  • Keywords

Nonlocal problem, Volterra integral, spectral element method, log orthogonal function, Legendre polynomial, weak singularity, exponential convergence.

  • AMS Subject Headings

65N35, 65M70, 41A25, 42B20, 45D05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-1041, author = {Zhang , ChaoYao , Guoqing and Chen , Sheng}, title = {A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {1041--1062}, abstract = {

In this paper, we propose a hybrid spectral method for a type of nonlocal problems, nonlinear Volterra integral equations (VIEs) of the second kind. The main idea is to use the shifted generalized Log orthogonal functions (GLOFs) as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals. This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions. The well-posedness and the related error estimates will be provided. Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0006s }, url = {http://global-sci.org/intro/article_detail/nmtma/21089.html} }
TY - JOUR T1 - A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels AU - Zhang , Chao AU - Yao , Guoqing AU - Chen , Sheng JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1041 EP - 1062 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0006s UR - https://global-sci.org/intro/article_detail/nmtma/21089.html KW - Nonlocal problem, Volterra integral, spectral element method, log orthogonal function, Legendre polynomial, weak singularity, exponential convergence. AB -

In this paper, we propose a hybrid spectral method for a type of nonlocal problems, nonlinear Volterra integral equations (VIEs) of the second kind. The main idea is to use the shifted generalized Log orthogonal functions (GLOFs) as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals. This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions. The well-posedness and the related error estimates will be provided. Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.

Zhang , ChaoYao , Guoqing and Chen , Sheng. (2022). A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 1041-1062. doi:10.4208/nmtma.OA-2022-0006s
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