New algorithms for approximating oscillatory Bessel integrals with Cauchy-type singularities
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 integr...
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| Vydáno v: | Results in applied mathematics Ročník 21; s. 100422 |
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Elsevier B.V
01.02.2024
Elsevier |
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| ISSN: | 2590-0374, 2590-0374 |
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| Abstract | 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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| AbstractList | 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. |
| ArticleNumber | 100422 |
| Author | Wu, Qinghua Sun, Mengjun |
| Author_xml | – sequence: 1 givenname: Qinghua surname: Wu fullname: Wu, Qinghua email: jackwqh@sina.com – sequence: 2 givenname: Mengjun surname: Sun fullname: Sun, Mengjun email: 459257638@qq.com |
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| Cites_doi | 10.1137/S0036142903431936 10.1016/j.cam.2021.113705 10.1007/BF01934465 10.1016/j.jmaa.2015.11.002 10.1093/imanum/drq035 10.1016/j.cam.2023.115220 10.1016/j.camwa.2016.05.016 10.1016/j.apnum.2012.01.009 10.1090/S0025-5718-1991-1068816-1 10.1016/j.jcp.2010.03.034 10.1016/j.apnum.2019.06.007 10.1155/2021/8021050 10.1007/s11075-015-0033-3 |
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| Keywords | 65D30 Bessel integral Complex integration method Gauss–Laguerre Cauchy-type singularities 65D32 |
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| SubjectTerms | Bessel integral Cauchy-type singularities Complex integration method Gauss–Laguerre |
| Title | New algorithms for approximating oscillatory Bessel integrals with Cauchy-type singularities |
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