A unified framework for asymptotic analysis and computation of finite Hankel transform

Wang, Haiyong

doi:10.1016/j.jmaa.2019.123640

Cited by 14 publications

(3 citation statements)

References 34 publications

(45 reference statements)

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“…This fundamental concept underwent further refinement by Iserles and Nørsett [5], who made a significant advancement by recognizing that the asymptotic expansion of b a f (x)e iωx dx solely depends on the values and derivatives of f (x) evaluated at the endpoints. Recently, this strategic approach has been extended to a broader spectrum of highly oscillatory integrals, including those with Besseltype kernels [16,20].…”

Section: The Filon-type Methods For I 1 [ F ]mentioning

confidence: 99%

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

Liu,

Xu,

2024

Mathematics

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In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫0bxαf(x)Ai(−ωx)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω≫1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+∞), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.

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Section: The Filon-type Methods For I 1 [ F ]mentioning

confidence: 99%

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

Liu,

Xu,

2024

Mathematics

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“…In the following tables, we make a comparison between the proposed method and the modified Filon-type method developed in [19]. Here, the multiplicity ℓ = 1 implies that we use Lagrange interpolation at the interpolation point.…”

Section: The Comparison With the Modified Filon-type Methodsmentioning

confidence: 99%

“…Regarding the computation of (1.1) with a special irregular oscillator and a = 0, Xiang and Wang in [25] first studied and gave an efficient Filon-type method by diffeomorphism transform for the special case g(a) = g ′ (a) = • • • = g (r) (a) = 0. Recently, Wang in [19] further comprehensively studied, provided asymptotic analysis and proposed the modified Filon-type method for computing (1.1) with the general oscillator g(x). It can achieve the higher accuracy by either adding derivatives at the critical points (including zeros, end points and stationary points) or increasing the number of interpolation nodes.…”

Section: Introductionmentioning

confidence: 99%

An Efficient and Accurate Numerical Method for the Bessel Transform with an Irregular Oscillator and Its Error Analysis

Wang

Kang

2023

Preprint

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In this paper we present and analyze an efficient numerical scheme for computing the Bessel transform with an irregular oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points for the general oscillator g(x), we derive a series of new quadrature formulae for such transform and carry out rigorous analysis for the proposed numerical method. The error analysis demonstrates that this method exhibits high asymptotic order, and the accuracy improves drastically with either increasing the frequency $\omega$ or adding more nodes. Compared with the existing modified Filon-type method, the established approach shows higher precision and error order at the same computational cost. The abundant numerical examples are provided to verify the theoretical results and illustrate the efficiency and accuracy of the proposed method. Mathematics Subject Classification (2000) 65D30 · 65D32 · 65R10 · 41A60

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Fast numerical integration of highly oscillatory Bessel transforms with a Cauchy type singular point and exotic oscillators

Kang,

Xu,

Liu

2024

Adv Comput Math

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A unified framework for asymptotic analysis and computation of finite Hankel transform

Cited by 14 publications

References 34 publications

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

An Efficient and Accurate Numerical Method for the Bessel Transform with an Irregular Oscillator and Its Error Analysis

Fast numerical integration of highly oscillatory Bessel transforms with a Cauchy type singular point and exotic oscillators

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