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Engineering with Computers
Published in: Engineering with Computers 4/2021

10-02-2020 | Original Article

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

Authors: Mahmoud A. Zaky, Ibrahem G. Ameen

Published in: Engineering with Computers | Issue 4/2021

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Abstract

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.
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Metadata
Title
A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions
Authors
Mahmoud A. Zaky
Ibrahem G. Ameen
Publication date
10-02-2020
Publisher
Springer London
Published in
Engineering with Computers / Issue 4/2021
Print ISSN: 0177-0667
Electronic ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-020-00953-9

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