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An accurate Legendre collocation method for third-kind Volterra integro-differential equations with non-smooth solutions

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Abstract

This work is to analyze a Legendre collocation approximation for third-kind Volterra integro-differential equations. The rigorous error analysis in the \(L^{\infty }\) and \(L_{\omega ^{0,0}}^{2}\)-norms is provided for the proposed method. In fact when converting the original equation to an equivalent second kind one, the integral operator of the obtained equation contains two singularities and may become non-compact under certain conditions. In addition, in order to avoid the low-order accuracy caused by the singularity of the solution at the initial point, we adopted the idea of smooth transformation at the beginning to convert the original equation into a new equation with a more regular solution. Finally, the validity and applicability of the method are verified by several numerical experiments.

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Acknowledgements

The authors would like to thank the referees for their valuable comments and helpful suggestions which led to an improved version of the paper.

Funding

This work was supported by NSF of China (Nos.11801127, 11771163, and 12011530058) and Scientific Research Foundation of Hunan Provincial Education Department (No.20C0081).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Changsha University of Science & Technology, Changsha, 410114, China

    Xiaohua Ma

  2. School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan, 430074, China

    Chengming Huang

  3. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science & Technology, Wuhan, 430074, China

    Chengming Huang

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  1. Xiaohua Ma
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  2. Chengming Huang
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Correspondence to Xiaohua Ma.

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Ma, X., Huang, C. An accurate Legendre collocation method for third-kind Volterra integro-differential equations with non-smooth solutions. Numer Algor 88, 1571–1593 (2021). https://doi.org/10.1007/s11075-021-01086-w

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  • DOI: https://doi.org/10.1007/s11075-021-01086-w

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