The generalized Euler-Maclaurin formula for the numerical solution of Abel-type integral equations
type: Item
instance of: scholarly article
title: The generalized Euler-Maclaurin formula for the numerical solution of Abel-type integral equations
zbMATH Open document ID: 1202.65180
DOI: 10.1216/JIE-2010-22-1-115
author: Johannes Tausch
published in: Journal of Integral Equations and Applications
publication date: 2010-05-26T00:00:00Z
review text: Based on an extension of the Euler-Maclaurin formula to singular integrals, introduced by \textit{I. Navot} [J. Math. Phys. 40, 271--276 (1961; Zbl 0103.28804)], the author derives a quadrature rule for integral equations of the Abel type. The author states that the method is particularly useful when combined with fast methods for evaluating integral operators. Discretization of the equation involves two sums. The first sum is the usual composite trapezoidal rule and the second sum can be considered to be a correction which accounts for the singularity of the integrand at \(t=\pi\). Section 2 is concerned with the generalised Euler-Maclaurin expansion. Brief details of a method involving the Mellin transform[\textit{P. Verlinden, A. Haegemans}, Numer. Math. 65, No. 3, 383--406 (1993; Zbl 0795.41029)] for integrals over a semi-infinite interval are given. Under the assumption that the solution takes a particular form an extension of the method to intervals of arbitrary length is discussed. The form of the remainder is proved in Lemma 2.1. A quadrature rule is derived and the form of the remainder in the equation by which the weights are determined is proved in Lemma 2.2. Section 3 concerns stability analysis. Results by \textit{G. H. Hardy} [Oxford: At the Clarendon Press (Geoffrey Cumberlege) XIV, 396 S. (1949; Zbl 0032.05801)], and by \textit{P. P. B. Eggermont} [J. Integral Equations 3, 317--332 (1981; Zbl 0469.65093)], are given and are used to prove Theorem 3.3. The latter gives information about the coefficients \(A_n\), the expansion coefficients of the reciprocal function. Two further lemmae are presented and proved before a necessary condition for stability is established in Lemma 3.6. Section 4 focuses on the discretization error. The quadrature error consists of two parts, one due to the truncation of the Euler-Maclaurin expansion and one due to replacing the derivatives of the expansion by finite differences. An asymptotic estimate for the discretization …
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 65B15
zbMATH DE Number: 5712449
zbMATH Keywords: Abel equations
zbMATH Keywords: Euler-Maclaurin formula
zbMATH Keywords: Volterra integral equations of the first kind
zbMATH Keywords: stability
zbMATH Keywords: convergence
reviewed by: Pat Lumb
MaRDI profile type: Publication
cites work: An Existence Theorem for Abel Integral Equations
cites work: Collocation Methods for Volterra Integral and Related Functional Differential Equations
cites work: Q3923560
cites work: Q5788558
cites work: Superconvergence of the iterated hybrid collocation method for weakly singular Volterra integral equations
cites work: Q3681889
cites work: High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equations
cites work: Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations
cites work: Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the First Kind
cites work: Numerical Quadrature and Asymptotic Expansions
cites work: The Euler-Maclaurin expansion and finite-part integrals
cites work: An Extension of the Euler-Maclaurin Summation Formula to Functions with a Branch Singularity
cites work: A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind
cites work: A fast method for solving the heat equation by layer potentials
cites work: An error expansion for cubature with an integrand with homogeneous boundary singularities
full work available at URL: jie-2010-22-1-115
OpenAlex ID: W2045065241
Recommended article: The Euler-Maclaurin expansion and finite-part integrals
Recommended article: Q4764047
Recommended article: Q3694637
Recommended article: Upgraded Gauss and Lobatto formulas wirh error estimation
Recommended article: Sharp integral inequalities based on general Euler two-point formulae
Recommended article: Q5080929
Recommended article: Compact numerical quadrature formulas for hypersingular integrals and integral equations
Recommended article: Numerical solution of the singular Abel integral equation by the generalized quadrature method
Recommended article: Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective
Recommended article: Q4863028
description: scientific article; zbMATH DE number 5712449