zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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= 0 u n , Nu= 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

34F05ODE with randomness
60H10Stochastic ordinary differential equations
34A34Nonlinear ODE and systems, general
35R60PDEs with randomness, stochastic PDE