Analytical Solution of Two Extended Model Equations for Shallow Water Waves by Adomian ’ s Decomposition Method

In this paper, we consider two extended model equations for shallow water waves. We use Adomian’s decomposition method (ADM) 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] 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 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 x s (5) However, Equation (3) can be transformed to the bilinear form where the solution of the equation is is given by the perturbation expansion where  is a bookkeeping non-small parameter, and where     are unknown functions that will be determined by substituting the last equation into the bilinear form and solving the resulting equations by equating different powers of  to zero.
The customary definition of the Hirota's bilinear operators are given by Also extended model of Equation ( 2) is obtained by the operator x D to the bilinear forms (4) and ( 5) 1 3 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 In a like manner, we extend Equation( 3) by adding the Using the properties of the D operators given above we obtain the extended model for Equation(3) given by In this paper, we use the Adomian's decomposition method (ADM) to obtain the solution of two considered equations above for shallow water waves.Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the ADM [4][5][6][7][8][9][10][11][12][13][14][15].A reliable modification of ADM has been done by Wazwaz [16].The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [4][5][6][7][8][9][10][11][12][13][14].This method efficiently works for initial-value or boundary-value problems and for linear or nonlinear, ordinary or partial differential equations and even for stochastic systems.Moreover, we have the advantage of a single global method for solving ordinary or partial differential equations as well as many types of other equations.

Basic idea of Adomian's Decomposition Method
We begin with the equation where L is the operator of the highest-ordered derivatives with respect to t and R is the remainder of the linear op-erator.The nonlinear term is represented by u .Thus we get The inverse  is assumed an integral operator given by The operating with the operator on both sides of Equation ( 18) we have f is the solution of homogeneous equation where involving the constants of integration.The integration constants involved in the solution of homogeneous equation (21) are to be determined by the initial or boundary condition according as the problem is initial-value problem or boundary-value problem.
The ADM assumes that the unknown function can be expressed by an infinite series of the form , , , n u 

ADM Implement for First Extended Model of Shallow Water Wave Equation
We consider the application of ADM to first extended model of shallow water wave equation.If Equation ( 14) is dealt with this method, it is formed as (25) where If the invertible operator is applied to Equation (25), then Copyright © 2011 SciRes. , is found.Here the main point is that the solution of the decomposition method is in the form of Substituting from Equation (29) in Equation( 28), we find , According to Equation (19) approximate solution can be obtained as follows: Thus the approximate solution for first extended model of shallow water wave equation is obtained as The terms , , , , , u x t u x t u x t in Eq ob (31), (32), (33).
d Extended

ADM Implement for Secon Model of Shallow Water Wave Equation
re we consider the application of ADM to second ex H tended model of shallow water wave equation.If Equation ( 16) is dealt with this method, it is formed as , , , If the invertible operator is obtained.By this is found.Here the main point is that the solution o 0 n (39) Substituting from Equation (39) in Eq fin , is found.
According to Equation ( 19) approximate solution can be obtained as follows: Thus the approximate solution for second extended model of shallow water wave equation is obtained as

Conclusions
In this paper, Adomian's decomposition method been successfully applied to find the solution of tw tended model equations for shallow water.The obtained results were showed graphically it is proved that Ado-    mian's decomposition method is a powerful method for solving these equations.In our work; we used the Maple Package to calculate the functions obtained from the Adomian's decomposition method.

Figure 1 .
Figure 1.For the first extended model of shallow water wave equation with the first initial condition (31) of Equation (14), ADM result for   u x t , , when c = 2.

Figure 2 .
Figure 2.For the second extended model of shallow waterwave equation with the first initial condition (31) of Equation (16), ADM result for   u x t , , when c = 2.
9) Some of the properties of the D-operators are as follows