Euler-Maclaurin expansions and approximations of hypersingular integrals

Chaolang Hu; Xiaoming He; Tao Lü

doi:10.3934/dcdsb.2015.20.1355

Article Contents

2015, Volume 20, Issue 5: 1355-1375. Doi: 10.3934/dcdsb.2015.20.1355

This issue Previous Article Error analysis for numerical formulation of particle filter Next Article A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential

Euler-Maclaurin expansions and approximations of hypersingular integrals

1.
College of Mathematics, Sichuan University, Chengdu,Sichuan, 610064
2.
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65401

Received: February 28, 2013

Revised: December 31, 2014

Published: June 2015

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Abstract

This article presents the Euler-Maclaurin expansions of the hypersingular integrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}% \frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and arbitrary non-negative integer $m$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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