Invariant Solutions using Symmetry Analysis with Conservation Laws for Various Nonlinear Partial Differential Equations Appearing in Physical Problems

Abstract

This thesis is dedicated to presenting a wide range of applications of continuous symmetry groups and conservation laws to the physically important system of partial differential equations. Lie symmetry analysis has been successfully implemented for obtaining exact analytic solutions, such as similarity solutions, fundamental solutions, wave-traveling solutions, series solutions, and soliton solutions of various partial differential equations. The main idea of the Lie group method is to utilize the invariance property of partial differential equations to obtain similarity reductions and group-invariant solutions. The invariance property is the most effective tool to find the symmetries of a differential equation because it enables us to reduce the number of independent variables by one. With the aid of this analysis, the system of partial differential equations has been reduced to a new system of ordinary differential equations, which brings an analytical solution to the main system. Infinitesimal generators, commutator tables, and group-invariant solutions have been carried out using the Lie symmetry approach. Also, a geometric approach to finding the symmetries of a system of partial differential equations has been talked about, in which Harrison and Estrabrook’s differential forms are used to construct the infinitesimal generators. However, there are an infinite number of possibilities for the linear combination of these symmetries, so to categorize the class of invariant solutions corresponding to these symmetries, an optimal system of subalgebra has been constructed using Olver’s standard approach. To broaden the range of symmetries of the system and therefore the family of solutions, nonlocal symmetries are introduced, in which the infinitesimals of the transformation must be dependent on the integration of local dependent variables. After that, the global behavior of dependent variables can be reflected by a nonlocal symmetry. An important part of the symmetry structure of a partial differential equation is information about its conservation laws. Knowledge of conservation laws for a partial differential equation provides insight into conserved physical quantities and can be used in the development of stable numerical methods. Therefore, in this work, conservation laws corresponding to the symmetries have been established by utilizing the fundamental approaches, namely, the direct approach, multiplier approach, and Ibragimov’s approach.