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Numerical steepest descent method for computing oscillatory-type Bessel integral transforms

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  • Chen, Ruyun
  • Li, Yu
  • Zhou, Yongxiong

Abstract

In this paper, numerical steepest descent method is implemented to approximate highly oscillatory Bessel-type integral transforms. We begin our analysis by utilizing an important relationship between Bessel function of the first kind and modified Bessel function of the second kind. Subsequently, we transform new integrals into the forms on the interval [0,+∞), where the integrands do not oscillate and decay exponentially fast. These integrals can then be efficiently computed using Gauss–Laguerre quadrature rule. Furthermore, we derive the theoretical error estimates that depend on the frequency ω and the number of nodes n. Numerical examples based on the theoretical results are provided to demonstrate the effectiveness of these methods.

Suggested Citation

  • Chen, Ruyun & Li, Yu & Zhou, Yongxiong, 2025. "Numerical steepest descent method for computing oscillatory-type Bessel integral transforms," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 236(C), pages 320-333.
    • Handle: RePEc:eee:matcom:v:236:y:2025:i:c:p:320-333
    DOI: 10.1016/j.matcom.2025.04.016
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    References listed on IDEAS

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    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. Wang, Hong & Kang, Hongchao & Ma, Junjie, 2024. "An efficient and accurate numerical method for the Bessel transform with an irregular oscillator and its error analysis," Applied Mathematics and Computation, Elsevier, vol. 473(C).
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