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 G G / -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. The KdV equation has infinitely many integrals of motion which do not change with time. we studied analytical solutions of two-component shallow water system and threecoupled KdV equations engendered by the Neumann system by Lie classical method. The symmetry group reduces the original equation to the simple one. By Lie classical method we (iv) 6 have investigated the symmetries. After having done standard change of dependent variables we used the basis of the infinitesimal operator written in the new variable in order to find some new classes of invariant solutions. Chapter 4 is concerned with nonlinear Klein-Gordon equation. The nonlinear KleinGordon equation appears in many types of nonlinearities. The Klein-Gordon equation arises in relativistic quantum mechanics and field theory, which is of great importance for the high energy physicists, and is used to model many different phenomena, including the propagation of dislocations in crystals and the behaviour of elementary particles. A numerical scheme has been developed for the numerical simulation of nonlinear Klein-Gordon equation. The essential difficulty in the numerical solutions for the Klein-Gordon equation involves the unboundedness of the physical domain, but in the chapter we considered the nonlinear KleinGordon equation on a bounded domain. Gauss-Lobatto-Chebyshev grid points are used for the numerical simulation. Chapter 5 deals with Fisher type’s equations. In this chapter, analytic and numerical solutions of nonlinear diffusion equation of Fisher type’s with the help of classical Lie symmetry method and differential qudrature method (DQM) are studied. The Fisher’s equation occurs in chemical kinetics and neutron population in a nuclear reaction. Moreover, the same equation also occurs in logistic population growth models, flame propagation, neurophysiology, autocatalytic chemical reactions, and branching Brownian motion processes. Lie symmetry method is utilized to investigate the symmetries and invariant solutions of the equations. The infinitesimal generators in the optimal system are used for reductions and exact solutions. Finally, polynomial differential qudrature method is used to find the numerical solutions of the Fisher’s type equations with the help of initial and boundary conditions taken from the analytic solutions obtained by classical Lie symmetry method. It is concluded that the numerical solutions are in good agreement with the analytical solutions. Chapter 6 is devoted to derive numerical solutions of two dimensional hyperbolic partial differential equation with variable coefficients. Hyperbolic PDEs describe propagation of disturbances in space and time when the total energy if the disturbances remain conserved. (v) 7 It’s the condition of energy conservation that makes the hyperbolic equation different from parabolic ones. In this chapter, cosine expansion based differential quadrature method extended for numerical simulation of the second order two dimensional hyperbolic equations. The main advantage of the present scheme is that the scheme gives very accurate and similar results to the exact solutions by choosing less number of grid points and the problem can be solved up to big time. The computed numerical results are compared with the results presented in some earlier works and it is found that the present numerical technique gives better results than the others.