An efficient numerical method for highly oscillatory logarithmic-algebraic singular integrals

SAIRA; Wenxiu Ma; Suliman Khan; SAIRA; Wenxiu Ma; Suliman Khan

doi:10.3934/math.2025224

AIMS Mathematics

2025, Volume 10, Issue 3: 4899-4914. doi: 10.3934/math.2025224

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Research article

An efficient numerical method for highly oscillatory logarithmic-algebraic singular integrals

1.
School of mathematical sciences, Zhejiang Normal University, Jinhua, 321004, China
2.
Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
3.
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
4.
School of Mathematical and Statistical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
5.
State Key Laboratory of Mechanics and Control for Aerospace Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

Received: 05 December 2024 Revised: 18 February 2025 Accepted: 26 February 2025 Published: 06 March 2025
MSC : 30E20, 32A55, 42B20, 45E05

This paper discussed the numerical evaluation of highly oscillatory integrals involving logarithmic and algebraic singularities. For an analytic function in a sufficiently large region containing $ [a, b] $, the integral was transformed into the sum of two line integrals where the integrands did not oscillate and decay exponentially. Thus, to approximate the line integrals, generalized Gauss-Laguerre quadrature and logarithmic Gauss-Laguerre quadrature were applied. The error bound and numerical results demonstrated that the proposed method efficiently obtained high-precision results even for high oscillations.
- algebraic singularities,
- logarithmic singularity,
- high oscillations,
- analytic function,
- Gauss-Laguerre quadrature
Citation: SAIRA, Wenxiu Ma, Suliman Khan. An efficient numerical method for highly oscillatory logarithmic-algebraic singular integrals[J]. AIMS Mathematics, 2025, 10(3): 4899-4914. doi: 10.3934/math.2025224

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Abstract

This paper discussed the numerical evaluation of highly oscillatory integrals involving logarithmic and algebraic singularities. For an analytic function in a sufficiently large region containing $ [a, b] $, the integral was transformed into the sum of two line integrals where the integrands did not oscillate and decay exponentially. Thus, to approximate the line integrals, generalized Gauss-Laguerre quadrature and logarithmic Gauss-Laguerre quadrature were applied. The error bound and numerical results demonstrated that the proposed method efficiently obtained high-precision results even for high oscillations.

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