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

Buyang Li, Weiwei Sun

Research output: Journal article publicationJournal articleAcademic researchpeer-review

12 Citations (Scopus)

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(hklnh|) for odd k and O(hk+1lnh) 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.
Original languageEnglish
Pages (from-to)1235-1255
Number of pages21
JournalIMA Journal of Numerical Analysis
Volume30
Issue number4
DOIs
Publication statusPublished - 1 Oct 2010
Externally publishedYes

Keywords

  • collocation
  • Hadamard finite-part integral
  • Newton-Cotes rule
  • superconvergence

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

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