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Efficient numerical methods for Cauchy principal value integrals with highly oscillatory integrands

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Abstract

This paper focus on the numerical evaluation of the Cauchy principal value integrals with oscillatory integrands where α, β > − 1,− 1 < τ < 1. For the case f is analytic in a sufficiently large region containing [− 1,1], the integrals can be transformed into the problems of integrating two line integrals, the integrands of which do not oscillate and decay exponentially fast, and thus can be computed by using Gaussian quadrature rules. For the smooth function f, a method is constructed by interpolating f at Clenshaw–Curtis points and the singular point τ, based on the fast computation of modified moments. Error bounds of two proposed methods are both presented. In addition, several numerical examples are given to illustrate the efficiency and accuracy of proposed methods.

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Acknowledgements

The authors are very grateful to the anonymous referees and the editors for their valuable suggestions and comments for great improvement of this paper.

Funding

This work was supported by the Youth Core Teachers Foundation of Zhengzhou University of Light Industry, and the National Natural Science Foundation of China (grant numbers 11701526, 11971446).

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Authors and Affiliations

  1. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, China

    Zhenhua Xu & Hongrui Geng

  2. School of Finance, Xuzhou University of Technology, Xuzhou, Jiangshu, 221018, China

    Zhanmei Lv

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  1. Zhenhua Xu
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  2. Zhanmei Lv
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  3. Hongrui Geng
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Correspondence to Zhenhua Xu.

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Xu, Z., Lv, Z. & Geng, H. Efficient numerical methods for Cauchy principal value integrals with highly oscillatory integrands. Numer Algor 91, 1287–1314 (2022). https://doi.org/10.1007/s11075-022-01302-1

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