Convergence of the polynomial projection method for the solution of ill-posed integrodifferential equations
type: Item
instance of: scholarly article
title: Convergence of the polynomial projection method for the solution of ill-posed integrodifferential equations
zbMATH Open document ID: 1163.65098
DOI: 10.3103/S1066369X07080014
author: Yu. R. Agachev
published in: Russian Mathematics
publication date: 2008-09-04T00:00:00Z
review text: The Cauchy problem \(x^{(j)}(-1)=0\), \(j\in\{0,...,m-1\}\), for the following integro-differential equation is considered: \((Gx)(t)+(Bx)(t)+(Hx)(t)=y(t)\), \(t\in [-1,1]\), where \((Gx)(t)=x^{(m)}(t)\), \((Bx)(t)=\sum_{k=1}^m g_k(t)x^{(m-k)}(t)\) and \((Hx)(t)=\sum_{j=0}^p\int_{-1}^1h_j(t,s)x^{(j)}(s)ds\), \(m<p\), and the functions \(y, g_k, h_j\) are known. This problem is ill-posed in standard (for differential equations) pairs of the spaces \((X,Y)\). The author chooses for \(X\) and \(Y\) appropriate Sobolev spaces with special weight functions; in this way, the Cauchy problem becomes well-posed in the sense of Hadamard. As a consequence, a method which uses projections onto subspaces of algebraic polynomials is shown to converge and convergence rates are obtained. In particular, the Galerkin method and the collocation method for the Cauchy problem are discussed.
reviewed by: Elena Resmerita
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 5321001
zbMATH Keywords: ill-posed problem
zbMATH Keywords: integro-differential equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: polynomial projection method
zbMATH Keywords: convergence
zbMATH Keywords: Galerkin method
zbMATH Keywords: collocation method
MaRDI profile type: Publication
cites work: Q5666993
cites work: Optimization of direct solution methods for ordinary integro-differential equations
cites work: Solution of one class of integro-differential equations by a mechanical quadrature method
cites work: Q5605345
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full work available at URL: s1066369x07080014
OpenAlex ID: W1996684697
description: scientific article; zbMATH DE number 5321001