A review of the decomposition method in applied mathematics
type: Item
instance of: scholarly article
title: A review of the decomposition method in applied mathematics
zbMATH Open document ID: 0671.34053
DOI: 10.1016/0022-247X(88)90170-9
author: G. Adomian
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The paper is a kind of selfreview. Let \(Fu=g\) be an ordinary nonlinear equation, \(F=L+R+N\), where L is ``easily invertible linear operator'', R is the remainder of the linear part of F, N is the nonlinearity. Then \(u=u_ 0+L^-Ru+L^-Nu,\) where \(Lu_ 0=0\). Write \(u=\sum^{\infty}_{0}u_ n\), \(Nu=\sum^{\infty}_{n=0}A_ n\), where \(\{A_ n\}\) are special polynomials, \(A_ n\) depends only on \(u_ 0,u_ 1,...,u_ n\). Then \(u_{n+1}=-L^-Ru_ n+L^{-1}A_ n\) and \(u_ n\) can be found successively. Polynomials \(A_ n\) should be constructed for each nonlinearity and the author proposes several formal schemes of such constructions which are the essence of the decomposition method by the author. He discusses the applications of this method to the systems of equations, stochastic equations, partial differential equations, considering for them both initial value problems and boundary problems. These applications are given in the numerous papers by the author and his colleagues (the list of references consists of 58 such papers). However the general or rigorous statements about convergence and error estimates are absent, although when numerical examples are considered, one can observe rather fast convergence, at least for fixed time. My opinion is that this formal method may happen to be a kind of variational method but its mathematical status is still not understood and justified.
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 4097854
zbMATH Keywords: decomposition method
zbMATH Keywords: stochastic equations
zbMATH Keywords: applications
zbMATH Keywords: error estimates
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
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full work available at URL: 0022-247x(88)90170-9
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