Lie group analysis and evolution of weak waves for certain hyperbolic system of partial differential equations

Abstract

In the present thesis, we study the applications of Lie group theory to system of quasilinear hyperbolic partial differential equations (PDEs), which are governed by many physical phenomena and having various important physical significance in the real life. Our primary objective in this thesis is to identify the symmetries of system of PDEs in order to obtain certain classes of group invariant solutions. The investigations carried out in this thesis
are confined to the applications of Lie group method to the system of quasilinear hyperbolic PDEs arising in magnetogasdynamics, two phase flows and other scientific fields. We organize the whole thesis into 7 chapters, described as follows. First chapter is introductory and deals with a short background history of Lie group of transformations and symmetries along with some of their important features which are of great importance in the work of proceeding chapters and the motivation behind our
interest. In the second chapter, we obtain exact solutions to the quasilinear system of PDEs, describing the one dimensional unsteady simple flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. Lie group of point transformations are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities. The next chapter deals with system of PDEs, governing the one dimensional unsteady
flow of inviscid and perfectly conducting compressible fluid in the presence of magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are
constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuity.