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A spectral collocation method for a weakly singular Volterra integral equation of the second kind

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Abstract

The solution y of a weakly singular Volterra equation of the second kind posed on the interval −1 ≤ t ≤ 1 has in general a certain singular behaviour near t = −1: typically, \(|y^{\prime }(t)| \sim (1+t)^{-\mu }\) for a parameter μ ∈ (0, 1). Various methods have been proposed for the numerical solution of these problems, but up to now there has been no analysis that takes into account this singularity when a spectral collocation method is applied directly to the problem. This gap in the literature is filled by the present paper.

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Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Fujian, 361005, China

    Can Huang

  2. Beijing Computational Science Research Center, Haidian District, Beijing, 100094, China

    Martin Stynes

Authors
  1. Can Huang
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  2. Martin Stynes
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Correspondence to Martin Stynes.

Additional information

Communicated by: I. Graham

The research of the first author is supported by the National Natural Science Foundation of China under grant 11401500 and the Fundamental Research Funds for the Central Universities under grant 20720150007. The research of the second author is supported in part by the National Natural Science Foundation of China under grant 91430216.

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Huang, C., Stynes, M. A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv Comput Math 42, 1015–1030 (2016). https://doi.org/10.1007/s10444-016-9451-6

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  • DOI: https://doi.org/10.1007/s10444-016-9451-6

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