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New modified Adomian decomposition method is proposed for the solution of the generalized fifth-order Korteweg-de Vries (GFKdV) equation. The numerical solutions are compared with the standard Adomian decomposition method and the exact solutions. The results are demonstrated which confirm the efficiency and applicability of the method.

The fifth-order (or generalized) KdV is the essential a model foe several physical phenomena including shallow-water waves near critical value of surface tension and waves in nonlinear LC circuit with mutual inductance between neighbouring inductors [

However, some of these methods are not easy to use and sometimes require tedious work and calculation [7, 8]. In recent years, Adomian decomposition methods (ADM) [

A comparative study between (ADM) and Crank Nicholas method presented in [

In 2001, Wazwaz presented another type of modification [

To the best of our knowledge, no attempt is made regarding the solution of generalized fifth order KdV equations by using modified decomposition method. So our main aim in this paper is used the modification ADM to solve the five particular class of the (gfKdV) equations. We generalized an appropriate Adomian’s polynomials for (gfKdV) equation will be handle more easily, quickly, and elegantly by implementing the new modified (ADM) rather than traditional methods for the exact solution of which is to be obtained subject to initial condition.

The well-known fifth-order KdV (fKdV) equations can be shown in the form

where a, b, c and d are nonzeros and real parameters, and is a sufficiently smooth function. The (fKdV) is an important mathematical model with wide applications in quantum mechanics and nonlinear optics.

Typical examples widely used in various fields such as solid state physics, plasma physics, fluid physics and quantum field theory. A variety of the (fKdV) equations can be developed by changing the real values of the parameters a, b and c [

1) The Sawada-Kotera (SK) equation is given by [

2) The Caudrey-Dodd-Gibbon (CDG) equation is given by [

3) The Lax equations [

4) The Kaup-Kuperschmidt (KK) equation [22,23]

5) The Ito equation [

In this section, we give outline and implement Adomian decomposition method for nonlinear equations to obtain analytic and approximate solutions which are obtained in a rapidly convergent series with elegantly computable components by this method. The Adomian approximation series converge quickly. In general, convergence regions of the series are small. Now we outline of the method here in order to obtain the solutions using (ADM), consider the fifth-order KdV Equation (1) in an operator form

where the notations and

By symbolizing the nonlinear term, respectively. The notation and symbolize the linear differential operators. Assuming the inverse of operator exists and it can conveniently be taken as the definite integral with respect to from 0 to, i.e.,

Thus, applying the inverse operator to (7) yields;

Therefore, it follows that

Since initial value is known and we decompose the unknown function as a sum of components defined by the decomposition series

with identified as.

The nonlinear terms and can be decomposed into infinite series of polynomial given by

and are the so-called Adomian polynomials of defined by

Substituting (11-14) into (10) gives rise to

The solution must satisfy the requirements imposed by the initial conditions. Based on the (ADM), we constructed the solution as

The decomposition method provides a reliable technique that requires less work if we compared with the traditional techniques.

In the new modification by Wazwaz [

A new recursive relationship expressed in the form

We can observe that algorithm (19) reduces the number of terms involved in each component, and hence the size of calculations is minimized compared to the standard (ADM) only. Moreover this reduction of terms in each component facilitates the construction of Adomian polynomials for nonlinear operators. the new modification overcomes the difficulty of decomposing and introduces an efficient algorithm that improves the performance of the standard (ADM).

In this section, we consider some (gfKdV) equations for numerical comparisons based on the new modifications of (ADM). In this paper, we illustrate how the approximate solutions of the (gfKdV) equations are close to exact solutions.

we consider the (S-K) Equation [

and the exact solution

we consider the (C-D-G) equation, with initial condition

and the exact solution

we consider Lax’s fifth order KdV equation with the initial condition:

and the exact solution

We consider the (K-K) equation with the initial condition

and the exact solution

we consider the Ito equation with the initial condition

and the exact solution

In this work, we proposed new modification of Adomian decomposition method. We solved the five well known forms of the (fKdV) equation with initial conditions. The method has been shown to computationally efficient in these examples that are important to researchers in applied sciences. The obtained results in examples indicted that the new modification of (ADM) was feasible and effective. The method overcomes the difficulties arising in the modified decomposition method established in [

The results show that the presented method is powerful mathematical tool for finding good approximate solu

_{0} = 0.0.

_{0} = 0.0.

_{0} = 0.0.

_{0} = 0.0.

tions of (gfKdV) equation with initial conditions and results are found to be in good agreement with the exact solution as shown from Figures 1-5. In addition, no linearization or perturbation is required by the method.