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Keywords = highly oscillatory Bessel kernel

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17 pages, 1144 KiB  
Article
Efficient Numerical Quadrature for Highly Oscillatory Integrals with Bessel Function Kernels
by Guo He and Yuying Liu
Mathematics 2025, 13(9), 1508; https://doi.org/10.3390/math13091508 - 3 May 2025
Viewed by 428
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 [...] Read more.
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. Full article
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17 pages, 551 KiB  
Article
Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels
by Chunhua Fang, Guo He and Shuhuang Xiang
Symmetry 2019, 11(2), 168; https://doi.org/10.3390/sym11020168 - 1 Feb 2019
Cited by 15 | Viewed by 3483
Abstract
In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The [...] Read more.
In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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15 pages, 484 KiB  
Article
On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels
by Junjie Ma and Huilan Liu
Symmetry 2018, 10(7), 239; https://doi.org/10.3390/sym10070239 - 25 Jun 2018
Cited by 11 | Viewed by 4981
Abstract
Lubich’s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the [...] Read more.
Lubich’s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the convolution quadrature rule for highly oscillatory problems. With the help of operational calculus, the convergence rate of the convolution quadrature rule with respect to the frequency is derived. Furthermore, its application to highly oscillatory integral equations is also investigated. Numerical results are presented to verify the effectiveness of the convolution quadrature rule in solving highly oscillatory problems. It is found from theoretical and numerical results that the convolution quadrature rule for solving highly oscillatory problems is efficient and high-potential. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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