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

Computational and Applied Mathematics, ISSN: 1807-0302, Vol: 41, Issue: 6
2022
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Article Description

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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