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

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(h k | ln h|) for odd k and O(h k+1 |ln h|) for even k when the singular point coincides with an element junction point. Two modified Newton-Cotes rules are proposed to remove the factor In 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 & Lu (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.