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IMA Journal of Numerical Analysis
Year: 2010  |  Volume: 30  |  Issue: 4  |  Page No.: 1235 - 1255

Newton-Cotes rules for Hadamard finite-part integrals on an interval

B Li and W. Sun    

Abstract:

The general (composite) Newton–Cotes rules are studied for Hadamard finite-part integrals. We prove that the error of the kth-order Newton–Cotes rule is O$$\left({h}^{k}\right|\mathrm{ln}h\left|\right)$$ for odd k and O$$\left({h}^{k+1}\right|\mathrm{ln}h\left|\right)$$ for even k when the singular point coincides with an element junction point. Two modified Newton–Cotes rules are proposed to remove the factor ln h from the error bound. The convergence rate (accuracy) of even-order Newton–Cotes rules at element junction points is the same as the superconvergence rate at certain Gaussian points as presented in Wu & Lü (2005, IMA J. Numer. Anal., 25, 253–263) and Wu & Sun (2008, Numer. Math., 109, 143–165). Based on the analysis, a class of collocation-type methods are proposed for solving integral equations with Hadamard finite-part kernels. The accuracy of the collocation method is the same as the accuracy of the proposed even-order Newton–Cotes rules. Several numerical examples are provided to illustrate the theoretical analysis.

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