How to solve nonlinear operator equation \(A(v^{2})+Cv=f\)
type: Item
instance of: scholarly article
title: How to solve nonlinear operator equation \(A(v^{2})+Cv=f\)
zbMATH Open document ID: 1065.47072
DOI: 10.1016/S0096-3003(03)00641-6
author: Zhong Chen
author: Minggen Cui
published in: Applied Mathematics and Computation
publication date: 2004-08-06T00:00:00Z
review text: In the function space \(X=W_2^1[a,b]\), the equation \(A(v^2)+Cv=f\) is studied, where \(A\) and \(C\) are bounded linear operators. Using a reproducing kernel, the nonlinear equation is reduced to a linear equation in \(W=X\otimes X\) whose solutions correspond to the solutions of the nonlinear equation if they have the form \(w(x,y)=v(x)v(y)\). This form is automatic under some uniqueness hypotheses which, however, are hard to verify. In order to construct a minimum norm solution of the linear equation (or, more general, of any linear equation in Hilbert spaces), the Gram-Schmidt orthogonalization process is applied for the adjoint equation. In some numerical examples, the error is small already after a few iterations. Some confusions of the fonts make it sometimes difficult to read the paper.
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 2082059
zbMATH Keywords: Gram-Schmidt process
zbMATH Keywords: reproducing kernel
zbMATH Keywords: linear equation
zbMATH Keywords: quadratic equation
MaRDI profile type: Publication
cites work: How to solve the equation \(AuBu+Cu=f\).
cites work: Q3743764
cites work: Theory of Reproducing Kernels
description: scientific article; zbMATH DE number 2082059