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Convergence analysis of a Nyström-type method for a class of nonlinear integral equations with highly oscillatory kernels

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  • Kassid, Qusay Abdulraheem
  • Sohrabi, Saeed
  • Ranjbar, Hamid

Abstract

In this paper, we present a Nyström-type method for the numerical solution of a class of nonlinear highly oscillatory Volterra integral equations with a trigonometric kernel. The implementation of this method leads to a nonlinear system that involves oscillatory integrals, which is then addressed using a two-point generalized quadrature rule to construct a fully discretized scheme. The error analysis of the method, in terms of both frequency and step length, is also presented. It is demonstrated that the proposed method outperforms the one recently introduced in the literature. To validate the method, several numerical examples are provided, confirming its efficiency and accuracy.

Suggested Citation

  • Kassid, Qusay Abdulraheem & Sohrabi, Saeed & Ranjbar, Hamid, 2025. "Convergence analysis of a Nyström-type method for a class of nonlinear integral equations with highly oscillatory kernels," Applied Mathematics and Computation, Elsevier, vol. 500(C).
    • Handle: RePEc:eee:apmaco:v:500:y:2025:i:c:s0096300325001778
    DOI: 10.1016/j.amc.2025.129450
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    References listed on IDEAS

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    1. Sohrabi, S. & Ranjbar, H. & Saei, M., 2017. "Convergence analysis of the Jacobi-collocation method for nonlinear weakly singular Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 299(C), pages 141-152.
    2. Zhao, Longbin & Wang, Pengde, 2022. "Error estimates of piecewise Hermite collocation method for highly oscillatory Volterra integral equation with Bessel kernel," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 137-150.
    3. S. Khudhair Abbas & S. Sohrabi & H. Ranjbar & Xian-Ming Gu, 2023. "Approximate Solution of a Class of Highly Oscillatory Integral Equations Using an Exponential Fitting Collocation Method," Journal of Mathematics, Hindawi, vol. 2023, pages 1-12, November.
    4. 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.
    5. Geng, F.Z. & Wu, X.Y., 2021. "Reproducing kernel function-based Filon and Levin methods for solving highly oscillatory integral," Applied Mathematics and Computation, Elsevier, vol. 397(C).
    6. 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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