The Research On Several Methods Of Nonlinear Partial Differential Equations

In nonlinear physics and mechanics, simplified nonlinear evolutions equations are often employed to describe the complex nonlinear physics system. The relations between physical quantities can be revealed by solving the nonlinear equations. Therefore, solving nonlinear equations is a key problem in nonlinear mechanics and physics. In recent years, with the development of symbolic computation accelerates the research of nonlinear partial differential equation has attained great developments. Many novel methods for the exact solutions of nonlinear partial differential equations are proposed.This dissertation mainly studies some aspects of nonlinear partial differential equations with the aid of symbolic computation, including the searching exact solutions of some nonlinear partial differential equations by the tanh methods and the exp-function method. Besides, based on the researches of many experts and scholars, a new method of solving nonlinear partial differential equations is proposed. Then, for some partial differential equations, we obtain a series of the exact solutions by using this method.First, introduce the basic theories of the tanh method and the exp-function method, and then with the aid of symbolic computation, we get rich types of exact solutions by using tanh method and the exp-function method for solving constant coefficients nonlinear partial differential equations and variable coefficient nonlinear partial differential equation which have practical application of background in physics and mechanics.Second, with a more in-depth research on nonlinear partial differential equation solutions, a new method for solving nonlinear partial differential equations is proposed based on f-expansion method, the exp-function method, the homogeneous balance method and (G'/G) expansion method. We refer to the proposed method as the (G'/(G+G')) expansion method. Then solve the classic KdV equation, Burgers equation, KdV-Burgers equation and the nonlinear Klein- Gordon equation by this method. The results show that it is an effective method for solving the nonlinear partial differential equations.Finally, take the (3+1)-dimensional Jimbo-Miwa equation and variable coefficient KdV equation as examples.Using(G'/(G+G')) expansion method, we get a series of exact solutions for the above equations. The results indicate that this method is concise and effective when dealing with the variable coefficient partial differential equation and the nonlinear partial differential equations which are complex and have practical significances.