STUDY OF SOME HYPERBOLIC SYSTEMS OF CONSERVATION LAWS AND WATER WAVE EQUATION

Abstract

Partial di erential equations have become signi cant tools to model and describe the fundamental theories arising in engineering science and physics. Researchers have developed a lot of systems to understand the behavior of natural phenomena in our surroundings. In the context of mathematical modeling, most of the physical phenomena in our surroundings are non-linear, hence modeled by non-linear partial di erential equations. To master the theory of partial di erential equations is a big challenge as we are not facilitated with a single theorem that is central to the subject here like in the theory of ordinary di erential equations, which relies on the fundamental \existence and uniqueness theorem". Instead of it, there exist various separate theories that can be used for particular types of partial di erential equations that arise in science very commonly. Even it is not always possible to have a solution in the case of nonlinear partial di erential equations. No single methodology has been developed to deal with all kinds of non-linear systems of partial di erential equations. Thus, to deal with these kinds of non-linear partial di erential equations, it is required to use special techniques to nd out their analytical solutions. Lie group analysis has become one of the signi cant tools to deal with the non-linear systems of PDEs. In the present thesis, we used the Lie group invariance property to study the non-ideal magnetogasdynamics and a higher-order non-linear partial di erential equation and obtained the solutions to the considered systems. The whole thesis contains ve chapters. A brief description of all the chapters is given as follows: