Recent Modification of Decomposition Method for Solving Nonlinear Partial Differential Equations

In this paper, efficient modification of Adomain decomposition method is proposed to solve nonlinear partial differential equations. Yields solution in rapid convergent series from easily computable terms to get exact solution, and yields in few iterations we get exact solution. Moreover, this modification does not require any linearization, discretization, or perturbations and therefore reduces the computations. Two illustration examples are introduced and illustrate the procedure of modification is simple yet highly accurate and rapidly converge to exact solution compares with the ADM or other modifications. The methodology presented here is useful for strongly nonlinear problems.


Introduction
In the last three decades, the Adomian Decomposition Method (ADM) has confirmed successful in getting analytical solution of non-linear differential equations by obtained solution in terms of convergent power series [1]. This method does not require discretization of the variables or domain [2,3]. The theoretical analysis of convergence for the series solution of ADM has been studied by Adomian [4], Cherrault [5], Cherrault et al. [6], and Chrysos et al. [7].
In recent years, there has been development in the application of ADM in solving partial differential equations (PDEs) with variable coefficients. For example, Wazwaz and Gorguis [8] also used the ADM solved linear heatlike and wave-like equations with variable coefficients. Soufyane and Boulmaf [9] applied ADM to get analytical solutions to the non-linear parabolic equation with variable physical parameters in time and space. Achouri and Omrani [10], applied the ADM to get numerical solutions for the damped gener alized regularized long-wave equation (DGRLW) with variable coefficients, and Tawfiq et al. [11][12][13][14] used this method for solving different model equations.
The aim of this paper is to get the exact analytical solution of nonlinear PDEs by using new efficient modification of ADM.

Suggested Modification of ADM
The procedure of the suggested modification for decomposition method (MADM) is introduced. The idea of modification is represented here by decomposed unknown function u (x, t) by infinite series as: Also, the nonlinear term ( , )can be represented by an infinite series of polynomials as follows: Then substitute equations (4) and (5) in (3) to obtain: Now, integration the right side, we have ) let m = m-1 on the right side of the above system, to get: From equation (7) am can be determining then substitute it in equation (4) to get the solution ( , ) .

Illustrative Applications
Here, the suggested modification will be used to solve the nonlinear PDE with initial condition as the following:

Example 1
Consider the following 3 rd order nonlinear PDEs: ℎ the nonlinear term defined as: Therefore, It is clear that the series is closed to This is exact solution.

The Convergence Analysis of Suggested Modification
In this section the convergence of the MADM will be discuss. Now, consider the general form of equation as: where H is the Hilbert space, N is the nonlinear operator N: H → H and f is also in H. Substituting the decomposition series equations (5) and (9)

Theorem 1
Let N be a nonlinear operator, N: H → H where H is Hilbert space and u be the exact solution of equation (10).   (10).

Conclusions
In this research we consider the homogeneous nonlinear PDEs with variable coefficients and initial condition. To solve it, new efficient modification of ADM is applied. Then, it is seen that suggested modification has the same exact solution. In addition to their effectiveness and usefulness in solving nonlinear PDEs, we show that this modification is powerful tools in solving nonlinear PDEs with initial conditions. Compared to ADM, other modification, or other methods for solving nonlinear PDEs, there is no need for linearization of nonlinear terms (16)(17), no need to calculate Adomian polynomial. Moreover, we can easily and rapidly attain the solution if we use MADM compared with ADM or other modifications as shown in solving illustrated examples.