IMA Journal of Numerical Analysis Advance Access published online on July 21, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp011
Newton–Cotes rules for Hadamard finite-part integrals on an interval
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Email: buyangli2{at}student.cityu.edu.hk
Corresponding author. Email: maweiw{at}math.cityu.edu.hk
Received on 20 June 2008. Accepted for publication 5 March 2009.
 |
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 
for odd k and 
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.
Key Words: Hadamard finite-part integral; Newton–Cotes rule; superconvergence; collocation