Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

Document type: Journal Articles
Article type: Original article
Peer reviewed: Yes
Full text:
Author(s): Nail Ibragimov, Ranis Ibragimov
Title: Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences
Journal: Mathematical Modelling of Natural Phenomena
Year: 2012
Volume: 7
Issue: 2
Pagination: 52-65
ISSN: 0973-5348
URI/DOI: 10.1051/mmnp/20127205
ISI number: 000303365000005
Organization: Blekinge Institute of Technology
Department: School of Engineering - Dept. of Mathematics & Natural Sciences (Sektionen för ingenjörsvetenskap - Avd.för matematik och naturvetenskap)
School of Engineering S-371 79 Karlskrona
+46 455 38 50 00
Language: English
Abstract: Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie's group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.
Subject: Mathematics\General
Keywords: internal gravity waves, invariant solutions