The Variational Iterational Method And The Approximate Solutions Of Some Nonlinear Equations

In the late 20th century, the nonlinear science is rapidly developing and become the frontier of science and technology. In the research of the nonlinear science, solving nonlinear equation is one of the most difficult and hottest topics. Solving soliton equations and integral equations are two of the most important fields of the nonlinear science. Many efficient methods for exploring the exact solutions of the soliton equations are presented by the mathematicians and physicists, such as inverse scattering method, Hirota bilinear methods, the homogeneous balance method, F-expansion method, exponential expansion method, etc. However, due to the variety and complexity of soliton equations, the only special soliton equations can be solved. Thus, looking for an effective approximate method of solving soliton equations and integral equations has become an important problem.Based on the Inokuti-Sekine-Mura's method, the variational iteration method is presented by Prof. He Ji-huan. Since the variational iteration method is introduced, more and more scientists have paid attention. As the rapid development of variational iteration method, it has been widely applied to the nonlinear problems. In this thesis, the approximate solutions of the soliton equation with high-order nonlinear terms, two coupled soliton equations are respectively presented with the aids of the variational iteration method and the approximate solutions are analyzed, the influence of coefficients of high-order nonlinear terms on the approximate solutions is discussed. And then the approximate solutions of FKPP equation are derived by variational iteration method. In the last, three two-dimensional integral equations are analyzed and the approximate solutions are obtained. The approximate solutions are simulated by Matlab.