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The numerical algorithms of infinite integrals involving products of Bessel functions of arbitrary order

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Abstract

The integrals of arbitrary products of zero-order and first-order Bessel functions (double Bessel functions) are widely used in practical problems. However, due to the strong oscillation and slow decay properties of the product terms of the double Bessel function, traditional numerical integration algorithms are no longer applicable. It is particularly important to develop high-precision and high-efficiency numerical algorithms for these types of integrals. In this paper, using the large argument approximate expression of the Bessel function and the accumulation property of the integral, double Bessel function integrals are transformed into the algebraic sum of ordinary integral forms in finite region and Fourier transform forms. The folded linear approach method and the Gaver–Stehfest (G–S) inverse Laplace transform method are developed for double Bessel function integrals and double-order single-argument Hankel transforms. Numerical examples are given to verify the efficiency and precision of these methods. The results show that the precision of the folded linear approach method is higher than that of the G–S inverse Laplace transform method, while the latter is more efficient and runs several times faster than the former. These two methods have different requirements for \(F(\lambda )\) when using the double Bessel function integrals. Therefore, the method that should be selected depends on the requirements of the specific \(F(\lambda )\) problem. This paper presents reliable computing methods for the study of fracture mechanics, electromagnetic responses, acoustic wave scattering, and droplet wetting.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant nos. 42064004 and 82060331), and the first class discipline construction project in Ningxia Universities: Mathematics.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

    Yonglin Yang, Shenghu Ding, Wenshuai Wang, Xu Wang & Xing Li

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  1. Yonglin Yang
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  2. Shenghu Ding
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  3. Wenshuai Wang
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  4. Xu Wang
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Correspondence to Wenshuai Wang or Xing Li.

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Communicated by Hui Liang.

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Yang, Y., Ding, S., Wang, W. et al. The numerical algorithms of infinite integrals involving products of Bessel functions of arbitrary order. Comp. Appl. Math. 41, 116 (2022). https://doi.org/10.1007/s40314-022-01816-3

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  • DOI: https://doi.org/10.1007/s40314-022-01816-3

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