NUMERICAL TREATMENT OF SOME PARTIAL DIFFERENTIAL EQUATIONS USING DIFFERENTIAL QUADRATURE METHOD

Abstract

The Differential Quadrature Method (DQM) is a numerical discretization technique for solving differential equations. The DQM, akin to the conventional integral quadrature method, approximates the derivative of a function at any location by a linear summation of all the functional values along a mesh line. The aim of the present thesis is to study some nonlinear partial differential equations using Differential Quadrature Method. In this thesis, numerical schemes are developed by using numerical techniques such as Quasilinearization Process, Differential Quadrature Method, Gauss Elimination Method and fourth-stage RK4 method [106]. The convergence and stability of the developed schemes have been studied. The Chapterwise summary of the thesis is as follows Chapter 1 is introductory in nature. Besides stating some numerical techniques like Finite Difference methods, Finite Element method, Finite Volume method and methods of weighted residuals it gives an introduction to Differential Quadrature Metho& and existing literature review. In Chapter 2, one-dimensional nonlinear Burgers' equation aU a2u .3u a<x<b, t>O (1) at ox2 a with initial and boundary conditions u(x,O)=q$(x), a<x<b and u(a,t)=f1(t), u(b,t)=f2(t), t>O is considered. In this Chapter, two different numerical schemes are proposed for the numerical solution of Burgers' equation (1). In first section, we have developed a numerical scheme based on differential quadrature method to solve time dependent Burgers' equation. In the construction of the numerical scheme, quasilinearization is used to tackle the nonlinearity of the problem which is followed by semi discretization for I spatial direction using differential quadrature method. Semi discretization of the problem leads to a system of first order initial value problem. For total discretization, we discretize the system of first order initial value problem resulting from the space semi-discretization using fourth-stage RK4 scheme with constant step length. The method is analyzed for stability. Finally, the method is illustrated and compared with existing methods via numerical experiment. The proposed numerical scheme is found to be quite easy to implement.