Applications of Discrete Symmetry Groups of Differential Equations

Abstract

This thesis is based on applications of the discrete point symmetry transformations of ordinary and partial di erential equations. First we provide a complete list of discrete symmetry transformations of third order scalar ordinary di erential equations by considering associated Lie algebras derived earlier. In this thesis, it is also presented that a discrete symmetry transformation of the heat equation improves the e ciency of Crank Nicolson method for the numerical solutions of the heat equation. The use of a discrete symmetry transformation to investigate solutions of an elastic wave equation is also a part of this thesis. It is shown that by considering a discrete symmetry transformations, new exact solutions of the elastic wave equation can be found. Thereby it is concluded that new exact solutions of a partial di erential equation by discrete symmetry transformations, can be generated. In this thesis, some new linear combinations of continuous symmetry transformations of the elastic wave equation are also considered which lead to many new solutions.