Applications of symmetry analysis to physically relevant differential equations.

Abstract

We investigate the role of Lie symmetries in generating solutions to differential equations that arise in particular physical systems. We first provide an overview of the Lie analysis and review the relevant symmetry analysis of differential equations in general. The Lie symmetries of some simple ordinary differential equations are found t. illustrate the general method. Then we study the properties of particular ordinary differential equations that arise in astrophysics and cosmology using the Lie analysis of differential equations. Firstly, a system of differential equations arising in the model of a relativistic star is generated and a governing nonlinear equation is obtained for a linear equation of state. A comprehensive symmetry analysis is performed on this equation. Secondly, a second order nonlinear ordinary differential equation arising in the model of the early universe is described and a detailed symmetry analysis of this equation is undertaken. Our objective in each case is to find explicit Lie symmetry generators that may help in analysing the model.