Analytical Solution of Two Extended Model Equations for Shallow Water Waves by He ’ s Variational Iteration Method

In this paper, we consider two extended model equations for shallow water waves. We use He’s variational iteration method (VIM) to solve them. It is proved that this method is a very good tool for shallow water wave equations and the obtained solutions are shown graphically.


Introduction
Clarkson et al. [1] investigated the generalized short water wave (GSWW) equation where  and  are non-zero constants.
Ablowitz et al. [2] studied the specific case 4   and 2   where Equation ( 1) is reduced to This equation was introduced as a model equation which reduces to the KdV equation in the long small amplitude limit [2,3].However, Hirota et al. [3] examined the model equation for shallow water waves 3 3 obtained by substituting 3     in (1).
Equation ( 2) can be transformed to the bilinear forms where s is an auxiliary variable, and f satisfies the bilinear equation However, Equation (3) can be transformed to the bi- where the solution of the equation is where f(x, t) is given by the perturbation expansion where ε is a bookkeeping non-small parameter, and Some of the properties of the D-operators are as follows   where Also extended model of Equation ( 2) is obtained by the operator 4 x D to the bilinear forms (4) and ( 5) where s is an auxiliary variable, and f satisfies the bilinear equation Using the properties of the D operators given above, and differentiating with respect to x we obtain the extended model for Equation (2) given by 4 In a like manner, we extend Equation ( 3) by adding the operator 4 x D to the bilinear forms ( 6) to obtain . 0 Using the properties of the D operators given above we obtain the extended model for Equation (3) given by 3 In this paper, we use the He's variational iteration method (VIM) to obtain the solution of two considered equations above for shallow water waves.The variational iteration method (VIM) [4][5][6][7][8][9][10] established in (1999) by He is thoroughly used by many researchers to handle linear and nonlinear models.The reliability of the method and the reduction in the size of computational domain gave this method a wider applicability.The method has been proved by many authors [11][12][13][14][15] to be reliable and efficient for a wide variety of scientific applications, linear and nonlinear as well.The method gives rapidly convergent successive approximations of the exact solution if such a solution exists.For concrete problems, a few numbers of approximations can be used for numerical purposes with high degree of accuracy.The VIM does not require specific transformations or nonlinear terms as required by some existing techniques.

Basic Idea of He's Variational Iteration Method
To clarify the basic ideas of VIM, we consider the following differential equation: where is a linear operator, a nonlinear operator and g t an inhomogeneous term.According to VIM, we can write down a correction functional as follows: where  is a general Lagrangian multiplier which can be identified optimally via the variational theory.The subscript indicates the th approximation and is considered as a restricted variation

VIM Implement for First Extended Model of Shallow Water Wave Equation
Now let us consider the application of VIM for first extended model of shallow water wave equation with the initial condition of: Its correction variational functional in x and t can be expressed, respectively, as follows: where  is general Lagrangian multiplier.
After some calculations, we obtain the following stationary conditions: Thus we have: As a result, we obtain the following iteration formula: We start with the initial approximation of   ,0 u x given by Equation ( 19).Using the above iteration formula (23), we can directly obtain the other components as follows: , d In Figure 1 we can see the 3-D result of first extended model of shallow water wave equation by VIM.

VIM Implement for Second Extended Model of Shallow Water Wave Equation
At last we consider the application of VIM for second extended model of shallow water wave equation with the initial condition given by Equation (19).Its correction variational functional in x and t can be expressed, respectively, as follows: where  is general Lagrangian multiplier.
After some calculations, we obtain the following stationary conditions:   Thus we have: As a result, we obtain the following iteration formula: We start with the initial approximation of   , 0 u x given by Equation (19).Using the above iteration formula (31), we can directly obtain the other components as follows: Copyright © 2011 SciRes.AJCM In Figure 2 we can see the 3-D result of second extended model of shallow water wave equation by VIM.

Acknowledgements
In this paper, He's variational iteration method has been successfully applied to find the solution of two extended model equations for shallow water.The obtained results were showed graphically it is proved that He's variational iteration method is a powerful method for solving these equations.In our work; we used the Maple Package to calculate the functions obtained from the He's variational iteration method.

Figure 1 .
Figure 1.For the first extended model of shallow water wave equation with the first initial condition (24) of Equation (14), when c = 2.

Figure 2 .
Figure 2.For the second extended model of shallow water wave equation with the first initial condition (32) of Equation (16), when c = 2.