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<p>This work presents an asymptotic analysis and efficient numerical schemes for approximating Fourier and Bessel integrals with a strong singularity and algebraic-logarithmic singularities. By employing the Cauchy integral theorem and Gaussian quadrature rules, specifically the generalized Gauss-Laguerre and logarithmic Gauss-Gautschi-Laguerre quadrature rules, we develop a modified numerical steepest descent method for two types of oscillatory hypersingular integrals. A rigorous error analysis is also conducted, and the corresponding convergence rates with respect to the frequency of oscillations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega\)</EquationSource> </InlineEquation>, are derived, indicating that the proposed numerical methods achieve higher accuracy as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\omega\)</EquationSource> </InlineEquation> increases. Several numerical experiments are provided to illustrate the effectiveness of the presented numerical approaches and their consistency with the theoretical analysis.</p>

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Asymptotic analysis and numerical evaluation of two types of oscillatory hypersingular integrals

  • 期刊论文

  • 发表日期：

    2025年12月2日

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  • Hong Wang,
  • Shuhuang Xiang,
  • Hongchao Kang

摘要

This work presents an asymptotic analysis and efficient numerical schemes for approximating Fourier and Bessel integrals with a strong singularity and algebraic-logarithmic singularities. By employing the Cauchy integral theorem and Gaussian quadrature rules, specifically the generalized Gauss-Laguerre and logarithmic Gauss-Gautschi-Laguerre quadrature rules, we develop a modified numerical steepest descent method for two types of oscillatory hypersingular integrals. A rigorous error analysis is also conducted, and the corresponding convergence rates with respect to the frequency of oscillations \(\omega\) , are derived, indicating that the proposed numerical methods achieve higher accuracy as \(\omega\) increases. Several numerical experiments are provided to illustrate the effectiveness of the presented numerical approaches and their consistency with the theoretical analysis.