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Fractional Collocation Method for Third-Kind Volterra Integral Equations with Nonsmooth Solutions

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Abstract

In this paper, we develop a collocation method for solving third-kind Volterra integral equations. In order to achieve high order convergence for problems with nonsmooth solutions, we construct a collocation scheme on a modified graded mesh using a basis of fractional polynomials, depending on a certain parameter \(\lambda \). For the proposed method, we derive an error estimate in the \(L^\infty \)-norm, which shows that the optimal order of global convergence can be obtained by choosing the appropriate parameter \(\lambda \) and modified mesh, even when the exact solution has low regularity. Numerical experiments confirm the theoretical results and illustrate the performance of the method.

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Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable suggestions and comments.

Funding

This work was supported by National Natural Science Foundation of China (Nos. 12171177 and 12011530058).

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Authors and Affiliations

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

    Zheng Ma & Chengming Huang

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

    Chengming Huang

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

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Ma, Z., Huang, C. Fractional Collocation Method for Third-Kind Volterra Integral Equations with Nonsmooth Solutions. J Sci Comput 95, 26 (2023). https://doi.org/10.1007/s10915-023-02155-z

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