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Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle

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Abstract

In this paper, the composite Newton–Cotes rule for evaluating hypersingular integral on a circle is investigated, we focus on the emphasis of the pointwise superconvergence. Taking the density function approximated, with the hypersingular kernel calculated analysis, we obtain the main error estimation of the general Newton–Cotes rule. We have proved that the property of the special function have big influence on the numerical results. We also present the relationship between the order of singularity and order of polynomial interpolation: if the hypersingular integral is \(p+1\) order, the order of polynomial interpolation must be \(p-1\) at least. The modified quadrature rules are proposed to improve the convergence rate which can reach the superconvergence rate. At last, we also presented some numerical examples to test our theoretical analysis.

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Acknowledgements

The work of ** Li was supported by Shandong Provincial Natural Science Foundation of China (Grant No. ZR2016JL006), Natural Science Foundation of Hebei Province (Grant No. A2019209533) and National Natural Science Foundation of China (Grant No.11771398). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

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  1. School of Science, Shandong Jianzhu University, **an, 250101, People’s Republic of China

    ** Li & Yongling Cheng

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  1. ** Li
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  2. Yongling Cheng
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Correspondence to ** Li.

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Communicated by Armin Iske.

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Li, J., Cheng, Y. Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle. Comp. Appl. Math. 41, 269 (2022). https://doi.org/10.1007/s40314-022-01951-x

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