Variational Iteration Method And Its Application To Nonlinear Equations

With the development of the computer science, there exists no difficulty in solving linear equations now, however, we have no a universal approach for nonlinear equations especially those with strong nonlinearity. The nonlinear science has been caught much attention since 1960s, and many different methods have been proposed to solve various different problems.In this thesis, we mainly study how to apply the variational iteration method to solve nonlinear equations. In Chapter 1, we give a brief introduction to nonlinear science and some methods for solving nonlinear equations, both the merits and disadvantages of those methods are briefly given. Chapter 2 presents the basic concepts of the variational iteration method, its new development and application. In order to illustrate its effectiveness and convenience, some examples are given. Finally comparison of the obtained approximate solutions with exact ones is made, revealing the convenience and effectiveness of the method. As an illustrating example, the K(m,n) equation is selected as an example in Chapter 3, which gives a detailed solution procedure of the variational iteration method to find exact solutions including compacton solution and solitary solution. Furthermore, we find that the compacton solution and the solitary solution can be transformed into each other. In Chapter 4, a new method called Exp-function method is suggested to seek the solutions of the nonlinear equation. We first we show its relation to the variational iteration method, then select the Liuville equation as an example to demonstrate the merits of the new method. We compare our solutions with those obtained by tanh method and Fan sub-equation method, revealing that our results include naturally those by tanh method and Fan sub-equation method as special cases.