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Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis

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Abstract

In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on the modification of the numerical steepest descent method, we present a new and efficient quadrature rule. Firstly, we divide the considered integrals into two parts by \(J_{m}(z)=\frac {1}{2}\left [H_{m}^{(1)}(z)+H_{m}^{(2)}(z)\right ]\), where each part can be transformed into the Fourier-type integrals. Then, we use the Cauchy’s residue theorem to convert these Fourier-type integrals into the infinite integrals on \([0,+\infty )\). Next, the resulting infinite integrals can be efficiently calculated by constructing some appropriate Gaussian quadrature rules. In addition, we conduct error analysis in inverse powers of the frequency parameter. Finally, several numerical examples are provided to show the efficiency and accuracy of the proposed method.

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Acknowledgments

The authors would like to express their most sincere thanks to the referees and editors for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Funding

This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010009, National Natural Science Foundation of China (Grant Nos. 11301125, 11971138), Research Foundation of Hangzhou Dianzi University (Grant No. KYS075613017)

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, People’s Republic of China

    Hongchao Kang, Chunzhi Xiang & Hong Wang

  2. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, People’s Republic of China

    Zhenhua Xu

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  1. Hongchao Kang
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  2. Chunzhi Xiang
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  4. Hong Wang
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Correspondence to Hongchao Kang.

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Kang, H., Xiang, C., Xu, Z. et al. Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis. Numer Algor 88, 1493–1521 (2021). https://doi.org/10.1007/s11075-021-01083-z

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