Compact numerical quadrature formulas for hypersingular integrals and integral equations
type: Item
instance of: scholarly article
title: Compact numerical quadrature formulas for hypersingular integrals and integral equations
zbMATH Open document ID: 1264.65033
author: Avram Sidi
published in: Journal of Scientific Computing
publication date: 2013-02-04T00:00:00Z
review text: Compact numerical quadrature formulas are developed for integrals having singularities at the end-points based on a novel technique recently created by the author using Euler-Maclaurin expansions. It is shown that the accuracy of the proposed quadrature formulas can be increased using the Richardson extrapolation process. The investigation includes the periodic hypersingular case providing a very high accuracy, the spectral accuracy. It is proved that the proposed quadrature formulas are exact for some periodic hypersingular integrals involving trigonometric polynomials. Moreover, it is proved that the accuracy becomes exponential when the integrands are analytic in a strip of the complex plane that includes the real axis. These quadrature formulas are applied to hypersingular integral equations both in periodic and nonperiodic case. The first part of this long and nice paper ends with a suitable numerical example that illustrates the accuracy of the quadrature formulas and confirm the presented theory. The second part of the paper is devoted to singular integrals having algebraic endpoints singularities extending the approach of the first part based on Euler-Maclaurin expansions. It is proved that the obtained quadratures are exact for some periodic Cauchy principal value integrals involving a family of trigonometric polynomials and the accuracy is proved to be spectral and it is illustrated on a numerical example involving periodic integrands.
reviewed by: Alexandru Mihai Bica
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 6134445
zbMATH Keywords: Hadamard finite part integral
zbMATH Keywords: hypersingular integrals
zbMATH Keywords: hypersingular integral equations
zbMATH Keywords: numerical quadrature
zbMATH Keywords: trapezoidal rule
zbMATH Keywords: midpoint rule
zbMATH Keywords: Euler-Maclaurin expansions
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: Richardson extrapolation
zbMATH Keywords: numerical example
zbMATH Keywords: Cauchy principal value integral
MaRDI profile type: Publication
full work available at URL: s10915-012-9610-y
OpenAlex ID: W1995272352
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DOI: 10.1007/S10915-012-9610-Y
description: scientific article; zbMATH DE number 6134445