Asymptotic decomposition of systems of nonlinear ordinary differential equations
type: Item
instance of: scholarly article
title: Asymptotic decomposition of systems of nonlinear ordinary differential equations
zbMATH Open document ID: 0545.34037
DOI: 10.1007/BF01057469
published in: Ukrainian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The construction of solutions of non linear ODE's by meams of asymptotic expansions relies on a relative-magnitude ordering of terms in the ODE's. This is most easily accomplished with the help of a 'small parameter'. Since usual small parameter expansions are not always successful, an alternate construction method is proposed. It relies on a special operator decomposition, in principle independent of any small parameter. Potential success is expected from the use of a variant of the 'similarity principle', the presence of similarity being associated with the presence of a transformation group. An operator decomposition with averaging scheme is worked out in terms of Lie series. The publication of a ''correctness''-proof is announced. The single illustrative example (Van der Pol equation) gives the impression that the scheme is quite laborious and cumbersome. Whether it provides a wider application the scope remains to be seen.
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 3867770
zbMATH Keywords: construction of solutions
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: small parameter
zbMATH Keywords: similarity principle
zbMATH Keywords: operator decomposition with averaging scheme
zbMATH Keywords: Lie series
zbMATH Keywords: Van der Pol equation
author: Yu. A. Mitropol'skij
author: Alexey K. Lopatin
reviewed by: Igor Gumowski
MaRDI profile type: Publication
cites work: Q4199088
cites work: Linear methods in problems of non-linear differential equations with a small parameter
cites work: Q5682682
cites work: Q3661941
cites work: Q4081425
full work available at URL: https://doi.org/10.1007/bf01057469
OpenAlex ID: W1983575311
description: scientific article; zbMATH DE number 3867770