Solution of singular integral equations by the method of oscillating functions
type: Item
instance of: scholarly article
title: Solution of singular integral equations by the method of oscillating functions
zbMATH Open document ID: 1187.65141
DOI: 10.3103/S1066369X09120044
author: L. B. Ermolaeva
published in: Russian Mathematics
publication date: 2010-01-12T00:00:00Z
review text: The following complete singular integral equation with Hilbert kernel is considered \[ Ax\equiv a(s)x(s)+\frac{b(s)}{2\pi} \int_{0}^{2\pi}\cot\frac{\sigma-s}{2}x(\sigma)d\sigma+ \frac{1}{2\pi}\int_{0}^{2\pi}h(s,\sigma)x(\sigma)d\sigma=y(s). \] Here \(x(s)\) is the unknown function, \(a(s), b(s), y(s)\) and \(h(s,\sigma)\) are known complex-valued \(2\pi\)-periodic functions. Moreover, \(a(s)\) and \(b(s)\) are assumed to be continuous, and \(y(s)\) and \(h(s,\sigma)\) are square summable in the \([0,2\pi]\) and \([0,2\pi]^2\) respectively. For the approximate solution of this equation the method of oscillating functions is proposed and theoretically justified. Also the author establishes the convergence rate of the method. In conclusion it is shown that the method of oscillating functions is order-optimal among all projective methods, as well as among all direct methods which allow one to construct an approximate solution in the form of polynomial \(x_n(s)=\sum_{k=-n}^{n}c_k e^{iks}\).
reviewed by: Alexander N. Tynda
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 5655841
zbMATH Keywords: singular integral equation
zbMATH Keywords: Hilbert kernel
zbMATH Keywords: uniform convergence
zbMATH Keywords: error estimates
zbMATH Keywords: order-optimal method
zbMATH Keywords: method of oscillating functions
zbMATH Keywords: projective methods
MaRDI profile type: Publication
cites work: Q5562285
cites work: Q3884682
cites work: Q5679291
cites work: Q3888485
cites work: Q4176512
cites work: Q3235550
Recommended article: Solution of ordinary integrodifferential equations by the method of oscillatory functions
Recommended article: Methods of solution of singular integral equations
Recommended article: The evaluation of the singular integrals of oscillating functions
Recommended article: A method for indefinite integration of oscillatory and singular functions
Recommended article: Solution of a class of singular integral equations
Recommended article: A Method for Solving Singular Integrodifferential Equations
Recommended article: Funtion-theoretic methods for singular integral equations
Recommended article: Oscillatory Singular Integrals
Recommended article: On the solution of a singular integral equation
Recommended article: Q4937789
description: scientific article; zbMATH DE number 5655841