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Singularity separation Chebyshev collocation method for weakly singular Volterra integral equations of the second kind

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Abstract

Volterra integral equation of the second kind with weakly singular kernel usually exhibits singular behavior at the origin, which deteriorates the accuracy of standard numerical methods. This paper develops a singularity separation Chebyshev collocation method to solve this kind of Volterra integral equation by splitting the interval into a singular subinterval and a regular one. In the singular subinterval, the general psi-series expansion for the solution about the origin or its Padé approximation is used to approximate the solution. In the regular subinterval, the Chebyshev collocation method is used to discretize the equation. The details of the implementation are also discussed. Specifically, a stable and fast recurrence procedure is derived to evaluate the singular weight integrals involving Chebyshev polynomials analytically. The convergence of the method is proved. We further extend the method to the nonlinear Volterra integral equation by using the Newton method. Three numerical examples are provided to show that the singularity separation Chebyshev collocation method in this paper can effectively solve linear and nonlinear weakly singular Volterra integral equations with high precision.

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Acknowledgements

The authors are very grateful to Editors and Referees for the valuable comments, which improve the quality of the paper significantly.

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. 11971241 and the Program for Innovative Research Team in Universities of Tianjin under Grant No. TD13-5078.

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  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Tongke Wang, Huan Lian & Lu Ji

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  1. Tongke Wang
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All authors contributed to the study conception and design. The manuscript was written by Tongke Wang. All authors read and approved the final manuscript.

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Wang, T., Lian, H. & Ji, L. Singularity separation Chebyshev collocation method for weakly singular Volterra integral equations of the second kind. Numer Algor 95, 1829–1854 (2024). https://doi.org/10.1007/s11075-023-01629-3

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