On the Coupling of Auxiliary Parameter, Adomian's Polynomials and Correction Functional

In this paper, we apply He's variational iteration method (VIM) coupled with an auxiliary parameter and Adomian's polynomials which proves very effective to control the convergence region of approximate solution. The proposed algorithm is tested on generalized Hirota–Satsuma coupled KdV equation and numerical results explicitly reveal the complete reliability, efficiency and accuracy of the suggested technique. It is observed that the approach may be implemented on other nonlinear models of physical nature.


Coupling Of Auxiliary Parameter And Correction Functional
An unknown auxiliary parameter h can be inserted into the correction functional (5) of He's VIM.According to this assumption, we construct the following variational iteration formula:  converge to the exact solution ) x ( u .It is the auxiliary parameter h which ensures that the assumption can be satisfied.In general, by means of the so-called h -curve, it is straightforward to choose a proper value of h which ensures that the approximate solutions are convergent.In fact, the proposed combination is very simple, easier to implement and is capable to approximate the solution more accurately in a bigger interval.

Coupling Of Auxiliary Parameter, Adomian's Polynomials And Correction Functional
In this algorithm, we will be making the coupling of Adomian's polynomials and the reformulated correctional functional (6) and obtain the following iterative scheme: where n A are the so-called Adomian's polynomials which can be generated according to the specific algorithms Abbasbandy [1,2].

NUMERICAL EXAMPLES
In this section, we apply the proposed algorithm (6a) to solve two generalized Hirota-Satsuma coupled KdV equations.Numerical results are compared with original variational iteration method (VIM).
Example 3.1 Consider the KdV equation (1) with the initial conditions (Wu [29], Yu [30], Yong [31])  , and  are arbitrary constants and the exact solutions are given by According to the original VIM, we have the following variational iteration formula (1): x xxx (9) Fig. 5 where n A , n B , n C , n D are the so-called Adomian's polynomials and can be generated by using the specific algorithm defined in Abbasbandy [1][2][3].First, to find the proper value of h for the approximate solutions (11), we plot the so-called h-curve of for the case 1 x  and 1 t  as shown in Fig. 1.According to these h-curves, it is easy to discover the valid region of h, which corresponds to the line segments nearly parallel to the horizontal axis.Here, we select 01 .0 h  . Fig.
 , and  are arbitrary constants and the exact solutions are by the original VIM is shown in Fig. 9 which confirms that the obtained results by original VIM is not valid for large values of x and t in example 3.2.Fig. 6.Absolute error for the 3rd-order Fig.

CONCLUSION
In this paper, we coupled an unknown auxiliary parameter and Adomian's polynomials in the correction functional of He's VIM for generalized Hirota-Satsuma coupled KdV equations.Numerical results and graphical representations explicitly reveal the complete reliability of this combination.It is observed that the used coupling can be very effective in solving complicated nonlinear problems of physical nature.
7. Absolute error for the 3th-order approximation by coupled form of VIM approximation by coupled form of VIM Fig.8.Absolute error for the 3rd-order Fig.9.Absolute error for the 3rd-order approximation by coupled form of VIM approximation by original VIM for