Integration of ODE with a small parameter via approximate symmetries: reduction of approximate symmetry algebra to a canonical form

Document type: Conference Presentations
Peer reviewed: Yes
Author(s): Rafail K. Gazizov, Nail H. Ibragimov, Veronika O. Lukashchuk
Title: Integration of ODE with a small parameter via approximate symmetries: reduction of approximate symmetry algebra to a canonical form
Conference name: International conference MOGRAN 11
Year: May 27-June 2, 2007
City: Karlskrona, Sweden
Organization: Blekinge Institute of Technology
Department: School of Engineering - Dept. Mathematics and Science (Sektionen för teknik – avd. för matematik och naturvetenskap)
School of Engineering S- 371 79 Karlskrona
+46 455 38 50 00
http://www.tek.bth.se/
Authors e-mail: nib@bth.se
Language: English
Abstract: The simplest method of integration of second-order differential equations using the Lie's canonical forms of two-dimensional algebras is well-known. We propose a generalization of this method on a case of integration of second-order differential equation with a small parameter having two approximate symmetries. The solution of such problem is reduced to the followings:
1) to classify approximate Lie algebras with two essential operators. As a result, seven different types of such Lie algebras have been obtained;
2) to construct canonical form of basic operators of non-similar algebras of every types for their realization in R2;
3) to set up general forms of invariant equations and formulas of their approximate solutions.
The similar problems are solved for systems of two ordinary differential equations with two approximate symmetries. On this way we have constructed representation of non-similar approximated Lie algebras in R3.
Subject: Mathematics\General
Keywords: Approximate symmetries, invariant equations, integration of second-order differential equation with a small parameter
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