A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions

被引：35

作者：

Zaky, Mahmoud A. ^{[1
]}

Ameen, Ibrahem G. ^{[2
]}

机构：

[1] Natl Res Ctr, Dept Appl Math, Cairo 12622, Egypt

[2] Al Azhar Univ, Fac Sci, Dept Math, Cairo, Egypt

来源：

ENGINEERING WITH COMPUTERS | 2021年 / 37卷 / 04期

关键词：

Spectral methods; Nonsmooth solution; Volterra integral equations; Weakly singular kernels; Convergence analysis; COLLOCATION METHODS; NUMERICAL-SOLUTIONS; CONVERGENCE; SUPERCONVERGENCE;

D O I：

10.1007/s00366-020-00953-9

中图分类号：

TP39 [计算机的应用];

学科分类号：

081203 ; 0835 ;

摘要：

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.

引用

页码：2623 / 2631

页数：9

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[3]

Brunner H., 2004, CAMBRIDGE MONOGRAPHS, V15

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