Exact and Numerical Solutions of Some Nonlinear Partial Differential Equations

Abstract

In this thesis, we established the range of applicability of Lie s classical symmetry method and differential quadrature method (DQM), with various of its generalizations, in constructing new exact and numerical solutions to the topical some nonlinear partial differential equations, including variable-coefficient Benjamin-Bona-Mahony-Burger (BBMB) equation, Coupled Short Pulse (CSP) equation with constant and variable coefficients, variable coefficients (2+1)-dimensional Diffusion-Advection (DA) equation, variable coefficients coupled KdV-Burgers equation, generalized Fitzhugh- Nagumo equation with time-dependent coefficients and two-space-dimensional Quasilinear Hyperbolic partial differential equation. Our thesis comprises of seven 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 2-7 Chapter 2 is concerned with variable-coefficients Benjamin-Bona-Mahony-Burger (BBMB) equation arising as mathematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The integrability of such an equation is studied with Painlevé analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Furthermore different types of solitary, periodic and kink waves can be seen with the change of variable coefficients. Chapter 3 deals with comparative study of travelling wave and numerical solutions for the Coupled Short Pulse (CSP) equation. The Lie symmetry analysis is performed Abstract v for coupled short plus equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator.