The First Integral Method For Solving Partial Differential Equations

As we all know, calculus was founded by two famous physicists and mathe-maticians Newton and Leibniz from17th century to18th century, and it has been widely applied in recent science research with increasing brilliant achievements, such as two-body problem and the discovery of Neptune are all great applications which shows that differential equations are successfully applied to particle dynam-ics. With the rapid development of analytical mechanics, people began to study continuous changes in the media, such as the law of the string vibration, elastic solid deformation, fluid dynamics, and other complex mechanical phenomena, and have got many partial differential models.In recent years, with the development of science and technology, the partial differential equations are widely used in meteorology, machinery, telecommunica-tions, chemical, ecological, economic, demographic and other social science fields. How to explain the dynamics and seek directly their explicit solutions of partial differential equations now become one of important issues for scientists.In this paper, by using of first integral methods, we investigated some new explicit solutions of partial differential equations. The first integral method was discovered by Z.S. Feng[1] for solving Burgers-KdV equation in2002, and it is an effective method of solving nonlinear partial differential equations. The first inte-gral method is based on ring exchange theory, by taking the traveling wave trans-formation in differential equations, we change partial differential equations into ordinary differential equations with a first, integral and find-the explicit solutions of ordinary differential equations, then obtain the solitary wave solutions of partial differential equations, the exponential function solutions, trigonometric function solutions and other exact solutions. Currently, the first integral method has been widely used to solve nonlinear partial differential equations. For example, by using the first integral method, Raslan[2], Abbasbandy and Shirzadi[3] and Tascanetal[4] obtained some exact solutions of Fisher equation, Benjamin-Bona-Mahony equa-tion and Zakharov-Kuznetsov equation and ZK-MEW equation, respectively. A significant advantage of the first integral method is to transform partial differential equations into ordinary differential equations with the first integrals.The thesis is divided into four chapters as follows. In the first chapter, we introduce the development and wide applications of partial differential equations, and the methods of solving exact solution of partial differential equations. We give, in the second chapter, the basie principle of the first integral method, and summarize the basic steps of the first integral for solving the exact solution of partial differential equations. In the third chapter, we will make full use of the first integral method to solve exact solutions of some important partial differential equations, including the generalized Whitham-Broer-Kaup(WBK) equation [5-6], a class of constant coefficient, of nonlinear partial differential equations [6-7], the generalized Drinfeld-Sokolov equations [8-9], a class of nonlinear partial differential equations [9-10], the generalized Mikhailov-Shabat(MS)[11-12] and the general-ized PHi-four equation [13-16]. Finally, in the fourth chapter, we summarizes the work in this paper and conclude future applications of the first integral method.