The Basic G G-Expansion Method for the Fourth Order Boussinesq Equation

The G G   -expansion method is simple and powerful mathematical tool for constructing traveling wave solutions of nonlinear evolution equations which arise in engineering sciences, mathematical physics and real time application fields. In this article, we have obtained exact traveling wave solutions of the nonlinear partial differential equation, namely, the fourth order Boussinesq equation involving parameters via the   G G  -expansion method. In this method, the general solution of the second order linear ordinary differential equation with constant coefficients is implemented. Further, the solitons and periodic solutions are described through three different families. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple.

Recently, Wang et al. [37] presented a widely used method to construct traveling wave solutions of different nonlinear partial differential equations (PDEs), called the basic G G  -expansion method.In addition, in this me- thod    G is executed as traveling wave solutions, where Later on, many researchers established exact traveling wave solutions of various nonlinear PDEs via this method.For example, Feng et al. [38] studied the well-known Kolmogorov-Petrovskii -Piskunov equation to obtain analytical solutions by using the same method.Naher et al. [39] constructed abundant traveling wave solutions of the higher-order Caudrey-Dodd-Gibon equation via this powerful method.In [40], Abazari and Abazari concerned about the same method for obtaining hyperbolic, trigonometric and rational function solutions of Hirota Many researchers studied the fourth order Boussinesq equation to construct analytical solutions by using different methods.For instance, Zhang [49] implemented the F-expansion method to investigate this equation for obtaining traveling wave solutions.In [50], Wazwaz employed the extended tanh method to construct exact solutions of the same equation.
The importance of this work is, the fourth order Boussinesq equation is considered to construct new exact traveling wave solutions including solitons, periodic and rational solutions by applying the basic G G  -expansion method.

  -Expansion Method
Suppose the general nonlinear partial differential equation:  , , , , , ,... 0, where is an unknown function, and the subscripts stand for the partial derivatives.

  ,t u x
The main steps of the basic G G   , -expansion method [37] are as follows: Step 1.Consider the traveling wave transformation: where H is the wave speed.Now using Equation (2), Equation ( 1) is converted into an ordinary differential equation for  : A  , ', , ,... 0, where the superscripts indicate the ordinary derivatives with respect to . Step 2. According to possibility, Equation ( 3) can be integrated term by term one or more times, yields constant(s) of integration.The integral constant may be zero, for simplicity.
Step 3. Suppose that the traveling wave solution of Equation ( 3) can be expressed in the form [37]: where  0,1, 2, , , and  are constants.
Step 4. To determine the integer n, substituting Equation (4) along with Equation (5) into Equation ( 3) and then taking the homogeneous balance between the highest order nonlinear terms and the highest order derivatives appearing in Equation (3).
Step 5. Substituting Equations ( 4) and ( 5  Then, substitute obtained values in Equation ( 4) along with Equation ( 5) with the value of n, we can obtain the traveling wave solutions of Equation (1).

Applications of the Method
In this section, the fourth order Boussinesq equation involving parameters is investigated to establish traveling wave solutions including three different families by

The Fourth Order Boussinesq Equation
In this work, we consider the fourth order Boussinesq equation followed by Wazwaz [50]: Boussinesq established Equation ( 6) to illustrate the propagation of long waves in shallow water.This equation also arises in other physical applications, for example, iron sound waves in plasma, nonlinear lattice waves and in vibrations in nonlinear string.Further, the details of this equation can be found in references [49][50][51][52][53][54].
Equation ( 6) is integrable, therefore, integrating with respect  twice and setting the constants of integration equal to zero, yields: Taking the homogeneous balance between 2  A and A in Equation ( 7), we obtain 2. n  Therefore, the solution of Equation ( 7) is of the form: where and are constants to be determined.
where , ,    and  are free parameters.
Substituting the general solution Equation (5) into Equation ( 8), we obtain three different families of traveling wave solutions of Equation ( 7 Family 3.1.3.Rational function solution: If U and V are taken as specific values, various known solutions can be rediscovered.

Trigonometric function solutions:
Substituting Equations ( 9) and ( 10) together with the general solution Equation ( 5) into the Equation ( 8), yields the trigonometric function solution Equation ( 12), we obtain following solutions respectively (if Also, substituting Equations ( 9) and ( 10) together with the general solution Equation ( 5) into the Equation ( 8), yields the trigonometric function solution Equation ( 12), our solutions become respectively (if but ): V Furthermore, substituting Equations ( 9) and ( 10) together with the general solution Equation ( 5) into the Equation ( 8), yields the trigonometric function solution Equation (12), our obtained traveling wave solutions become respectively (if 0,

Rational function solutions:
Substituting Equations ( 9) and ( 10) together with the general solution Equation ( 5) into the Equation ( 8), we obtain the rational function solution Equation ( 13), and our wave solutions become respectively (if 2

Results and Discussion
It is worth declaring that some of our obtained solutions are in good agreement with already published results which are presented in Table 1.Moreover, some of obtained exact solutions are described in Figures 1-12.
Beyond this table, we obtain new exact traveling wave solutions A 1 , A 3 , A 5 , A 6 , A 7 , A 9 , A 11 , A 12 , A 13 , and A 14 ,          which are not being established in the previous literature.

Conclusion
In this article, the basic   G G  -expansion method with second order linear ordinary differential equation is implemented to investigate the fourth order Boussinesq equation involving parameters.The obtained solutions are presented through the hyperbolic functions, the trigonometric functions and the rational functions.Further, it is important mentioning that some of our solutions are coincided with published results and others have not been stated in earlier literature.The solutions show that this method is reliable and straightforward solution method to obtain exact solutions of nonlinear evolution equations.Therefore, this powerful method can be effectively used to construct new exact traveling wave solutions of diffe-

Step 6 .
Solve the system of algebraic equations which are obtained in Step 5 with the aid of algebraic software Maple and we obtain values for  

0 1 2
Substituting Equation(8) together with Equation (5) into the Equation (7), the left-hand side of Equation (7) is converted into a polynomial of , According to Step 5, collecting all terms with the same power of   G G  .After that, setting each coefficient of the resulted polynomial to zero, yields a set of algebraic equations (for simplicity, which are not presented) for with the help of algebraic software Maple, we obtain two different values.