Approximate Solutions For Some Nonlinear Partial Differential Equations

In the field of natural science and social science, many phenomena can be described by nonlinear equations. Therefore, the research on these problems is equivalent to the study of nonlinear equations. In recent years, due to the rapid development of computer symbolic computing software, solving linear equations has become simpler than before. However, it is difficult to solve the nonlinear equations. Therefore, the research on approximate solution of nonlinear equations has become a hot issue. Many approaches have been proposed to solve the approximate solution of nonlinear equations recently. Among them, the homotopy perturbation method and the homotopy analysis method are two important methods.In this paper, we use the homotopy analysis method to study the nonlinear partial differential equations with constant coefficients, using the homotopy perturbation method and Flourier transformation to solve the nonlinear partial differential equations with variable coefficients, and obtain the approximate solutions. Taking the Drinfel’d-Sokolov-Wilson equation as an example, we obtain two different forms of approximate solutions when we select the basis functions in the form of trigonometric function and exponential function. Similarly,when we take the basis function in the form of exponential function, we get the approximate solution of the exponential form of the generalized Boussinesq equation. The error analysis is carried out by adjusting the auxiliary parameter η in the approximate solution, so as to select the proper auxiliary parameter to ensure the convergence of the solution, and finally ensure the validity of the approximate solution. This shows that it is a good method to solve this kind of the nonlinear partial differential equations with constant coefficients by using the homotopy analysis method. However, it is difficult to solve the nonlinear partial differential equation with variable coefficients. In the study of the variable coefficient Boussinesq equation and the variable coefficient KdV-Burgers equation, we use the homotopy perturbation method and get the two order approximate solutions in the different forms.