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The generalized quadrature method for a class of highly oscillatory Volterra integral equations

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Abstract

A generalized quadrature method is studied for Volterra integral equations with highly oscillatory kernels. According to the kernel, a two-point quadrature rule is constructed by Lagrange’s identity at first. The error of the quadrature formula is presented as well. Then, it is employed to discretize the highly oscillatory equation without the need to compute moment. For the convergence, the asymptotic order as well as the classical order of the quadrature method for equation is analyzed. It is shown that the method has asymptotic order two and converges with order two as step length decreases. Some numerical examples are conducted to test its efficiency.

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Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments and suggestions which lead to an improvement of this paper.

Funding

This work was supported by NSF of China (Nos. 12171177, 11771163) and the Fundamental Research Funds for the Universities of Henan Province (No. NSFRF220409).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China

    Longbin Zhao

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

    Chengming Huang

  3. 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. Longbin Zhao
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  2. Chengming Huang
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Correspondence to Chengming Huang.

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Zhao, L., Huang, C. The generalized quadrature method for a class of highly oscillatory Volterra integral equations. Numer Algor 92, 1503–1516 (2023). https://doi.org/10.1007/s11075-022-01350-7

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