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An hp-version fractional collocation method for Volterra integro-differential equations with weakly singular kernels

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Abstract

In this paper, we propose an \(\varvec{hp}\)-version fractional collocation method for solving second kind Volterra integro-differential equations with weakly singular kernels. We derive \(\varvec{hp}\)-version error estimates in a weighted \(\varvec{H}^{1}\)-norm for the method on arbitrary meshes. The results show that for any given mesh partition, exponential rates of convergence can be achieved for certain weakly singular solutions by linearly increasing the degrees of piecewise fractional polynomials. The results also imply that in the case of uniform mesh, the method has no (\(\varvec{h}\)-version) order barrier for weakly singular solutions, which is different from classical polynomial collocation methods. The method is easy to implement and has the same computational complexity as polynomial collocation methods. Numerical experiments are presented to demonstrate the efficiency of the proposed method.

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Acknowledgements

The authors are very grateful to the anonymous referees and the editors for their valuable suggestions and comments.

Funding

This work was supported by the 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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Correspondence to Chengming Huang.

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Ma, Z., Huang, C. An hp-version fractional collocation method for Volterra integro-differential equations with weakly singular kernels. Numer Algor 92, 2377–2404 (2023). https://doi.org/10.1007/s11075-022-01394-9

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  • DOI: https://doi.org/10.1007/s11075-022-01394-9

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