Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

N. H. Ibragimov; R. N. Ibragimov

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 2, page 52-65
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Ibragimov, N. H., and Ibragimov, R. N.. "Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences." Mathematical Modelling of Natural Phenomena 7.2 (2012): 52-65. <http://eudml.org/doc/222193>.

@article{Ibragimov2012,
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.},
author = {Ibragimov, N. H., Ibragimov, R. N.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {internal gravity waves; invariant solutions},
language = {eng},
month = {2},
number = {2},
pages = {52-65},
publisher = {EDP Sciences},
title = {Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences},
url = {http://eudml.org/doc/222193},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Ibragimov, N. H.
AU - Ibragimov, R. N.
TI - Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/2//
PB - EDP Sciences
VL - 7
IS - 2
SP - 52
EP - 65
AB - 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.
LA - eng
KW - internal gravity waves; invariant solutions
UR - http://eudml.org/doc/222193
ER -

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