Corrington's Walsh function method applied to a nonlinear integral equation
type: Item
instance of: scholarly article
title: Corrington's Walsh function method applied to a nonlinear integral equation
zbMATH Open document ID: 0816.65138
author: B. G. Sloss
author: William F. Blyth
published in: Journal of Integral Equations and Applications
publication date: 1995-02-21T00:00:00Z
review text: The authors consider the integral equation \[ u(x) = \sum^ N_{i = 1} b_ i(a_ i(x) + \int^ x_ 0 k_ i(x,t) u(t) dt)^ i\tag{*} \] for \(x \overline{\in} I := [0,1]\), where \(b_ i \overline {\in} \mathbb{R}\), \(a_ i \overline{\in} L^ \infty(I)\) and \(k_ i \overline{\in} L^ \infty(I \times I)\). By the use of Banach's fixed point theorem the existence of a unique solution \(u \overline {\in} L^ 2\) of (*) in the ball \(B_ c := \{w \overline{\in} L^ 2 : \| w\|_ 2 \leq c\}\) with \(c\) sufficiently small is proven. The solution is approximated by finite series of Walsh-functions (special orthonormal functions). This approximated solution is obtained by projecting relation (*) onto a finite-dimensional span of Walsh-functions and applying the usual iteration method on this equation. This is the method of \textit{M. S. Corrington} [A solution of differential and integral equations with Walsh functions. IEEE Trans. Circuit Theory, CT-20, No. 5, 470-476 (1973)]. Error estimates for the approximated solution are derived on certain regularity conditions. [Remark: (1) The fixed point theorem of Banach implies existence and uniqueness of the solution. Therefore, the proof of the uniqueness in Theorem 1 is superfluous. (2) A linear Volterra integral equation with a continuous kernel has no characteristic value. Hence, the assumptions of Theorem 3 are empty, i.e., there exists no bifurcation point \((b_ 0,0)\)].
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 716120
zbMATH Keywords: nonlinear Volterra integral equation
zbMATH Keywords: error estimates
zbMATH Keywords: finite series of Walsh-functions
zbMATH Keywords: iteration method
reviewed by: Walter Petry
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075806
OpenAlex ID: W1989314466
cites work: Q5558293
cites work: Q3999467
cites work: A Remarkable Series of Orthogonal Functions (I)
cites work: A Remarkable Series of Orthogonal Functions (II)
cites work: Q3997391
cites work: A priori error estimates for Corrington's Walsh function method
DOI: 10.1216/JIEA/1181075806
description: scientific article; zbMATH DE number 716120