Solutions of some nonlinear partial differential equations

Abstract

The prime objective and motivation in carrying out the proposed study is to demonstrate the importance and efficacy of group theoretic techniques and differential qudrature method to find analytical and numerical solutions of some nonlinear equations viz. family of nonlinear equations, two component shallow water wave system and three-coupled KdV equations, Klein-Gordon equation, Fisher s type equations and two dimensional hyperbolic equations with variable coefficients. Our thesis comprises of six chapters. In the introductory part some important features of Lie group of transformations and differential quadrature method (DQM) are demonstrated and the mathematical fundamentals of continuous group theory and weighting coefficients of DQM are reviewed which are of great importance to the work dealt in Chapters 2-6. Chapter 2 is concerned with family of nonlinear evolution equations. In mathematical physics the family of nonlinear evolution equations has been a subject of extensive study. It represents a class of nonlinear evolution equations. We investigate the symmetry of the equation by means of classical Lie symmetry method. The symmetry algebras and groups of family of nonlinear equation are obtained. Specially, the most general one-parameter group of symmetries is given and most general solutions are gained. We used and#61480;G Gand#61602; / and#61481;-expansion method to find travelling wave solutions of one ODE. New explicit solutions of equation are derived. Chapter 3 deals with two-component shallow water system and three-coupled KdV equations engendered by the Neumann system by using Lie classical method. The shallow water equations are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid. KdV is a mathematical model of waves on shallow water surface.