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A review of the decomposition method in applied mathematics. (English) Zbl 0671.34053

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.
Reviewer: L.Pastur

MSC:

34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34A34 Nonlinear ordinary differential equations and systems
35R60 PDEs with randomness, stochastic partial differential equations
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References:

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