Algebraic Method For Solving Modified Boussinesq Equation

The exact traveling wave solution is one of most important solutions during solving the nonlinear partial differential equations. Traveling waves, whether their solutions are in explicit or implicit forms, are very interesting form the point of view of applications. These types of waves will not change their shapes during propagation and are thus easy to detect, of particular interesting are three types of traveling waves: the solitary waves, which are localized traveling waves, asymptotically zero at large distances; the periodic waves; the kink waves, which rise or decent from one asymptotic state to another. Recently, there many methods for obtain the exact traveling wave solutions of a nonlinear partial differential equation. Some of the most important methods, for instance, tanh-function method, nonlinear transformation method, sine-cosine method, trial function method and so on. However, using these methods does not obtain the periodic wave solutions of the nonlinear wave equations, only get the solutions of the solitary wave and the kink wave solutions. This paper will firstly introduce and apply tanh-function method and extended tanh-function method to obtain the traveling wave solutions of the Boussinesq equation; secondly, we will introduce and apply Jacobi elliptic function expansion method get the traveling wave solutions of the Boussinesq equation. Finally we introduce and apply a unified algebraic method called the mapping method to obtain exact traveling wave solutions for a large variety of nonlinear partial differential equations. This method includes several direct methods as special case, such as tanh-function method, sech-function method and Jacobi elliptic function method. Above all, by means of this method, the solitary wave, the periodic wave and the kink wave solution can, if they exist, be obtained simultaneously to the equation in question without extra efforts. A large variety of exact traveling solutions of nonlinear partial differential equations are obtained by means of the mapping method, the modified mapping method and the extended mapping method, as long as odd-and even-order derivative terms do not coexist in nonlinear partial differential equations under consideration.