Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

Abstract

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.

Conservation laws and invariant solutions of the non-linear governing equations associated with a thermodynamic model of a rotating detonation engines with Korobeinikov׳s chemical source term

“…The detailed review of this subject can be found e.g. in [11] and in [4] and here we emphasize that part of the review in the latter references that is essential in the frame of the current work.…”

Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets

“…be the velocity vector, p(x, y, z, t) be the pressure and ρ(x, y, z, t) be the density in the Cartesian coordinates (x, y, z). The starting model describing the large-scale geophysical motions for baroclinic jets that release potential energy in the atmosphere are given by nonlinear model for internal waves under the Boussinesq approximation ( [20], [21], [22], [23]):…”

Two approximate symmetry frameworks for nonlinear partial differential equations with a small parameter: Comparisons, relations, approximate solutions

“…At the first step towards the classification of symmetries of a differential equation-based model involving arbitrary elements, it is essential to find equivalence transformations for this model. Equivalence transformations map the given DE system to another differential equation from the same general class [26, 30]. Lie groups of equivalence transformations can be sought systematically using techniques similar to those used to determine admitted Lie symmetry groups.…”