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Quadrature formulae of many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities and their error analysis

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  • Kang, Hongchao
  • Xu, Qi

Abstract

In this article, we propose and analyze the Clenshaw–Curtis–Filon-type method for computing many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities at the endpoints. First, by using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. Numerical experiments verify the efficacy and correctness of the presented method.

Suggested Citation

  • Kang, Hongchao & Xu, Qi, 2023. "Quadrature formulae of many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities and their error analysis," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    • Handle: RePEc:eee:apmaco:v:442:y:2023:i:c:s0096300322008268
    DOI: 10.1016/j.amc.2022.127758
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    References listed on IDEAS

    as
    1. He, Guo & Zhang, Chuanlin, 2017. "On the numerical approximation for Fourier-type highly oscillatory integrals with Gauss-type quadrature rules," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 96-104.
    2. Kang, Hongchao & An, Congpei, 2015. "Differentiation formulas of some hypergeometric functions with respect to all parameters," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 454-464.
    3. 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).
    Full references (including those not matched with items on IDEAS)

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