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Clenshaw–Curtis-type quadrature rule for hypersingular integrals with highly oscillatory kernels

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  • Liu, Guidong
  • Xiang, Shuhuang

Abstract

The Clenshaw–Curtis-type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points ▪ for any smooth functions g(x). Based on the fast Hermite interpolation, this paper provides a stable recurrence relation for these modified moments. Convergence rates with respect to the frequency k and the number of interpolation points N are considered. These theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:340:y:2019:i:c:p:251-267
    DOI: 10.1016/j.amc.2018.08.004
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    References listed on IDEAS

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    1. Xu, Zhenhua & Milovanović, Gradimir V. & Xiang, Shuhuang, 2015. "Efficient computation of highly oscillatory integrals with Hankel kernel," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 312-322.
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    Cited by:

    1. Xu, Zhenhua & Geng, Hongrui & Fang, Chunhua, 2020. "Asymptotics and numerical approximation of highly oscillatory Hilbert transforms," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    2. SAIRA & Shuhuang Xiang & Guidong Liu, 2019. "Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function," Mathematics, MDPI, vol. 7(10), pages 1-11, September.
    3. Li, Bin & Xiang, Shuhuang, 2019. "Efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    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. SAIRA & Wen-Xiu Ma, 2022. "An Approximation Method to Compute Highly Oscillatory Singular Fredholm Integro-Differential Equations," Mathematics, MDPI, vol. 10(19), pages 1-16, October.
    6. Liu, Guidong & Xiang, Shuhuang, 2023. "An efficient quadrature rule for weakly and strongly singular integrals," Applied Mathematics and Computation, Elsevier, vol. 447(C).
    7. Kang, Hongchao & Wang, Ruoxia & Zhang, Meijuan & Xiang, Chunzhi, 2023. "Efficient and accurate quadrature methods of Fourier integrals with a special oscillator and weak singularities," Applied Mathematics and Computation, Elsevier, vol. 440(C).

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