Study of Some Nonlinear Partial Differential Equations for Lie Symmetries and Exact Solutions

Abstract

The objective of the thesis entitled, Study of Some Nonlinear Partial Differential Equations for Lie Symmetries and Exact Solutions , is to study the applications of Lie groups to the nonlinear partial differential equations (NLPDEs). The prime objective in this newlinethesis is to examine the Lie symmetries of the NLPDEs in order to obtain the exact solutions, which are helpful in demonstrating the integrability and physical behavior of the nonlinear equations. newlineDuring the last few decades, investigations of exact solutions to nonlinear partial differential equations have played a vital role in the study of physical phenomena. Exact solutions provide the proper understanding of qualitative features of many nonlinear physical phenomena and processes in various areas of natural science. newline In recent years, the generalization of the constant coefficients to variable coefficients has grown predominantly in research interest. Because the differential equations with variable coefficients characterize many nonlinear phenomena more realistically than equations with constant coefficients, but often, it is difficult to solve explicitly these nonlinear differential equations for exact solutions. However, there is much current interest in finding the exact explicit solutions of these nonlinear equations. The exact solutions provide much information about physical phenomena and various other aspects of these nonlinear systems. The exact solutions are also helpful to examine and discuss the sensitivity of physical phenomena with several important parameters described by variable coefficients. These solutions are also helpful in designing and testing numerical algorithms. newlineMathematical methods which generate a wide range of explicit solutions and applicable to all type of nonlinear differential equations are few. The group-theoretic techniques can be categorized in this class, and generally, it produces a variety of exact solutions, directly or via similarity solutions, classifying invariant equations and/or reducing the number of indep