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. In all types, we find reductions in terms of system of ordinary differential equations and exact solutions of the CSP equation are derived which are compared with numerical solutions using classical fourth order Runge-Kutta (RK4) scheme. Using the Lie symmetry approach, we have examined herein the variant Coupled Short Pulse (VCSP) equation. The method reduced the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations in all optimal subgroups. The system of ordinary differential equations is further solved in general to obtain exact and numerical solutions. Several new physically important families of exact and numerical solutions are derived. Chapter 4 is concerned with (2+1)-dimensional variable coefficients Diffusion- Advection (DA) equation, which arises in modeling of various physical phenomena, is studied by Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation and then exact and numerical solutions are reported for the reduced second order nonlinear ordinary differential equations. Further, an extended G /G -expansion method is applied to DA equation for constructing some new non-traveling wave solutions. In chapter 5, we apply Lie symmetry approach, modified G /G -expansion method and classical RK4 method for seeking the exact and numerical solutions of variable coefficients coupled KdV-Burgers’ equation. Using suitable similarity transformations, the given system of partial differential equations reduced to a system of ordinary differential equations (ODEs). Moreover, the system of ODEs are solved by the modified G /G -expansion method to obtain exact solutions. Further, RK4 method is applied to system of ODEs for constructing numerical solutions of variable coefficients coupled KdV-Burgers’ equation. Chapter 6 is devoted to derive the numerical solutions of the generalized Fitzhugh- Nagumo equation with time-dependent coefficients in one space dimension are considered using polynomial differential quadrature method (PDQM). The PDQM reduced the problem into a system of second order linear differential equation. Then, Abstract vi the obtained system is solved by optimal four-stage, order three strong stabilitypreserving time-stepping Runge-Kutta (SSP-RK43) scheme. The accuracy of the proposed method is demonstrated by three test examples. The numerical results are shown in max absolute errors ( ),  L root mean square errors (RMS) and relative errors ( ) 2 L form. Numerical solutions obtained by this method when compared with the exact solutions reveal that the obtained solutions produce high accurate results. Numerical results show that the proposed method is of high accuracy and is efficient for solving the generalized Fitzhugh-Nagumo equation. In Chapter 7, we have proposed a numerical technique based on PDQM to find the numerical solutions of two-space-dimensional Quasilinear Hyperbolic partial differential equations subject to appropriate dirichlet and neumann boundary conditions. The second-order hyperbolic partial differential equations have great impotence in fluid dynamics and aerodynamics, theory of elasticity, optics, electromagnetic etc. The PDQM reduced the equations into a system of second order linear differential equation. The obtained system is solved by RK4 method by converting into a system of first order ordinary differential equations. The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions. The proposed technique can be applied easily for multidimensional problems.