NUMERICAL SOLUTION OF SOME NON-LINEAR DIFFERENTIAL EQUATIONS

Abstract

Most of the practical problems in physics, engineering and other applied sciences can be formulated in the form of differential equations. These differential equations could be ordinary or partial. The theory of partial differential equations has long been one of the most important fields in Mathematics. In general, analytical methods for the solution of these equations are preferred, as they lead to general rather than particular solutions. Moreover, analytical solutions give more information. However, there are several situations in which a numerical method may be preferred despite an analytical solution being available. Unfortunately engineering and science problems are generally highly complex, often involving nonlinear phenomena and analytical solutions are not always possible to find. With much interest and great demand for the solution of-nonlinear partial differential equations, several numerical methods have been proposed, for example the finite difference, finite element, boundary element methods etc. In this thesis an attempt has been made to develop a numerical method to solve nonlinear evolutionary partial differential equations. This method exploits the finitely reproducing property of nonlinear operator of the equation. A nonlinear operator M is called finitely reproducing with respect to a complete orthonormal set of functions , [oklic.i, if for newthere exists rn(n)E N, such that m(n) M[ 5(0 = (0,042(t),...,an(0) Ok(x) j7=1 k=1 where gk's, k=1,2,...,m(n) are explicitly known functions of als, j=1,2,...,n. In the proposed numerical method a nonlinear partial differential equation is reduced to a set of nonlinear ordinary differential equations (ODE's) by using finitely reproducing property of the nonlinear term in the equation. This system of nonlinear ODE's is solved by the s-stage Runge-Kutta-Chebyshev second order method. The solution of original partial differential equation is obtained by using the solution of the system of ODE's. The advantages of the method are that the solution can be obtained at any point in the space domain directly and it does not involve dicey matrix computations. The validity of the method is tested by comparing the results obtained with exact or previously published results. A very good agreement is found. In this thesis, well known equations like Burgers, Kuramoto-Sivashinsky, Schrbdinger, Korteweg-de Vries, regularized long wave, Fermi-Pasta-Ulam and "good" Boussinesq equations which have widespread applications in several branches of engineering and applied sciences, are solved. The thesis comprises of the following eight chapters, The chapter I of the thesis is Introduction. In this chapter a brief description of nonlinear partial differential equations considered in the thesis is given along with a survey of literature on the existing numerical methods for the solution of these equations. The proposed method is also described in detail. In chapter II, the numerical solution of the Burgers equation ut — c uxx + uu ----= 0 a=x-sb, t>0 where c is a parameter, is presented. This equation arises in the study of shock waves, turbulence problems and continuous stochastic processes. This chapter is divided into two sections. In section A, the non-periodic boundary conditions are considered. The nonlinear term in the equation is shown to be finitely reproducing with respect to basis functions (577,, siaork.i. The numerical method is derived for both homogeneous and nonhomogeneous boundary conditions_ The presented method is applied on four examples of (iv) different kinds including a propagating shock problem and shock wave approaching a steady state. Computed results are presented graphically and in the form of tables. The section B deals with finding the numerical solution of Burgers equation with periodic boundary conditions. It is shown that the nonlinear term is finitely reproducing with co [V(2n) 1 eirttv respect to the basis functions = -CO Three example problems are solved for situations of fixed shock front and moving shock front. Results obtained are compared with exact and previously published results and a very good agreement is found. In chapter III, the numerical treatment of the Kuramoto-Sivashinsky (KS) equation ut 4- uux + du + v u = 0 asx < , t > 0 xx where d and v are parameters, is presented. The KS equation describes a weakly nonlinear evolution of the instability at the interface between two viscous fluids flowing concurrently. The nonlinear term in this equation is same as in Burgers equation and is proved finitely reproducing like in section B of chapter II. The effect of nonlinearity (by varying d and v) on the solutions are shown through graphs for two examples for different initial shapes of interface. In chapter IV, numerical experience with the nonlinear Schrodinger equation ILL+ cr u 4- alul2t4 -= 0 xx " nr -1 , 12=-1 where q is a real parameter, is reported. The analytical properties of the nonlinear Schrodinger equation are well known. It is one of the relatively small number of soliton producing equations. The chapter is divided into two sections. In section A, the Schrodinger equation with non-periodic boundary conditions is considered. The nonlinear term in the equation is shown to be finitely reproducing with respect to basis functions (EWE sin(jx)). • A single soliton and two soliton propagation problems are solved as J=1 example problems and results are given in the form of three dimensional figures for different values of q. In other example, the values of two conserved quantities, namely, (v) u I 2dx and .1( I ux12— lq I u I4)dx are computed. In this section the Ginzburg-Landau equation of which SchrOdinger equation is a particular case is also solved for one example problem. In section B of this chapter method is applied to the Schrodinger equation with periodic boundary conditions. The nonlinear term is proved to be finitely {I i mr e 03 reproducing with respect to basis functions . The values of the conserved 427-ir quantities are given and compared with exact and previously published results for two example problems. In section A of chapter V, the numerical solution of the Korteweg-de Vries (KdV) equation ut + c uux+ cuxxx = 0 0<x <L, t>0 with periodic boundary conditions using finitely reproducing property of the nonlinear term, is presented. The KdV equation plays a major role in the study of nonlinear dispersive waves. One set of solutions of KdV equation is a family of solitary waves. Four example problems, three exhibiting soliton behavior and one non-soliton behavior are solved. Computed results are presented graphically. The errors in the conserved quantities, namely, fu2dx and f I u3— ux2 rdz are also given. The obtained results are found to be comparable with previously published results. In Section B the regularized long wave (RLW) equation ut + uux— aunt = 0 t>0 introduced as an alternative model to the KdV equation, is solved. The word regularized refers to the fact that RLW equation has certain technical advantages over the KdV equation. Two example problems are solved in which one is two solitons collision problem and the second has source function in the right hand side of the equation. Computed errors are given in tabular form and other results are presented graphically.