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Mathematical Problems in Engineering
Volume 2015, Article ID 918083, 9 pages
http://dx.doi.org/10.1155/2015/918083
Research Article

The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

Jin Li1,2

1School of Science, Shandong Jianzhu University, Jinan 250101, China
2School of Mathematics, Shandong University, Jinan 250100, China

Received 27 July 2015; Accepted 20 September 2015

Academic Editor: Kishin Sadarangani

Copyright © 2015 Jin Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.