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Efficient Numerical Quadrature for Highly Oscillatory Integrals with Bessel Function Kernels

Authors

Guo He, Yuying Liu
Type: Article
Publication Date: 2025-05-03
Citations: 0
DOI: https://doi.org/10.3390/math13091508

Abstract

In this paper, we investigate efficient numerical methods for highly oscillatory integrals with Bessel function kernels over finite and infinite domains. Initially, we decompose the two types of integrals into the sum of two integrals. For one of these integrals, we reformulate the Bessel function Jν(z) as a linear combination of the modified Bessel function of the second kind Kν(z), subsequently transforming it into a line integral over an infinite interval on the complex plane. This transformation allows for efficient approximation using the Cauchy residue theorem and appropriate Gaussian quadrature rules. For the other integral, we achieve efficient computation by integrating special functions with Gaussian quadrature rules. Furthermore, we conduct an error analysis of the proposed methods and validate their effectiveness through numerical experiments. The proposed methods are applicable for any real number ν and require only the first ⌊ν⌋ derivatives of f at 0, rendering them more efficient than existing methods that typically necessitate higher-order derivatives.

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Fast computation of highly oscillatory Bessel transforms

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This study focuses on the efficient and precise computation of Bessel transforms, defined as ∫abf(x)Jν(ωx)dx. Exploiting the integral representation of Jν(ωx), these Bessel transformations are reformulated into the oscillatory integrals … This study focuses on the efficient and precise computation of Bessel transforms, defined as ∫abf(x)Jν(ωx)dx. Exploiting the integral representation of Jν(ωx), these Bessel transformations are reformulated into the oscillatory integrals of Fourier-type. When a>0, these Fourier-type integrals are transformed through distinct complex integration paths for cases with b<+∞ and b=+∞. Subsequently, we approximate these integrals using the generalized Gauss–Laguerre rule and provide error estimates. This approach is further extended to situations where a=0 by partitioning the integral's interval into two separate subintervals. Several numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed algorithms.

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In this paper, we present an efficient numerical algorithm for approximating integrals involving highly oscillatory Bessel functions with Cauchy-type singularities. By employing the technique of complex line integration, the highly … In this paper, we present an efficient numerical algorithm for approximating integrals involving highly oscillatory Bessel functions with Cauchy-type singularities. By employing the technique of complex line integration, the highly oscillatory Bessel integrals are transformed into oscillatory integrals with a Fourier kernel. When the integration interval does not contain zeros, we use Cauchy's theorem to transform the integration path to the complex plane and then use the Gaussian–Laguerre formula to compute the integral. For cases in which the integration interval contains zeros, we decompose the integral into two parts: the ordinary and the singular integral. We give a stable recursive formula based on Chebyshev polynomials and Bessel functions for ordinary integrals. For singular integrals, we utilize the MeijerG function for efficient computation. Numerical examples verify the effectiveness of the new algorithm and the fast convergence.

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