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Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

Author

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  • Jianyu Wang

    (College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China
    These authors contributed equally to this work.)

  • Chunhua Fang

    (College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China
    These authors contributed equally to this work.)

  • Guifeng Zhang

    (College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China)

Abstract

In this paper, we focus on the numerical solution of the second kind of Volterra integral equation with a highly oscillatory Fourier kernel. Based on the calculation of the modified moments, we propose four collocation methods to solve the equations: direct linear interpolation, direct higher order interpolation, direct Hermite interpolation and piecewise Hermite interpolation. These four methods are simple to construct and can quickly compute highly oscillatory integrals involving Fourier functions. We present the corresponding error analysis and it is easy to see that, in some cases, our proposed method has a fast convergence rate in solving such equations. In some cases, our proposed methods have significant advantages over the existing methods. Some numerical experiments demonstrating the efficiency of the four methods are also presented.

Suggested Citation

  • Jianyu Wang & Chunhua Fang & Guifeng Zhang, 2023. "Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels," Mathematics, MDPI, vol. 11(20), pages 1-19, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4249-:d:1257674
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    References listed on IDEAS

    as
    1. Liu, Guidong & Xiang, Shuhuang, 2019. "Clenshaw–Curtis-type quadrature rule for hypersingular integrals with highly oscillatory kernels," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 251-267.
    2. Tuan, Nguyen Huy & Mohammadi, Hakimeh & Rezapour, Shahram, 2020. "A mathematical model for COVID-19 transmission by using the Caputo fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. Yang, Xuehua & Wu, Lijiao & Zhang, Haixiang, 2023. "A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity," Applied Mathematics and Computation, Elsevier, vol. 457(C).
    4. Hao Chen & Ling Liu & Junjie Ma, 2020. "Analysis of Generalized Multistep Collocation Solutions for Oscillatory Volterra Integral Equations," Mathematics, MDPI, vol. 8(11), pages 1-16, November.
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