An Approximate Solution to The Newell-Whitehead Equation by Adomian Decomposition Method

The decomposition method was first introduced by Adomian since the beginning of the 1980s [7]. The Adomian method [3] can be used for solving a wide range of problems whose mathematical models yield equation or system of equations involving algebraic, differential, integral and integro-differential. In this method the solution is considered as the sum of an infinite series, rapidly converging to an accurate solution. It is well known that the key of the method is to decompose the nonlinear term in the equations into a peculiar series of polynomials 1 n n A ∞


Introduction
The decomposition method was first introduced by Adomian since the beginning of the 1980s [7]. The Adomian method [3] can be used for solving a wide range of problems whose mathematical models yield equation or system of equations involving algebraic, differential, integral and integro-differential. In this method the solution is considered as the sum of an infinite series, rapidly converging to an accurate solution.
It is well known that the key of the method is to decompose the nonlinear term in the equations into a peculiar series of polynomials  [4][5][6].
This iterative method has been proven to be rather successful in dealing with linear problems as well as nonlinear. Adomian gives the solution as an infinite series usually converging to an accurate solution.
An Analytical Solution of the Stochastic Navier-Stokes System is shown by Adomian [3]. The decomposition method used to solve a system of partial differential equations and in reaction-diffusion to the Brusselator model and finding that the Adomian series solution gives an excellent approximation to the exact solution [20]. Wazwaz [22] developed a fast and accurate algorithm for the solution of sixth-order boundary value problems (BVP) and the modified decomposition method [19,21]. Abbasbandy [1,2] and Allan [8] studied some efficient numerical algorithms to solve a system of two nonlinear equations (with two variables) based on Newton's method and a numerical solution of the Blasius equation. Wang [18] presented a new algorithm for solving the classical Blasius equation.
Hashim [11] studied the Adomian decomposition method for solving BVPs for fourth-order integro-differential equations showing that with a few modifications the Adomian's method can be used to obtain the known results of the special functions of mathematical physics and the Blasius equation [13].
Kechil et al. [15] applied a non perturbative solution of free-convective boundary-layer equation by ADM. Chang [9] presented a decomposition solution for fins with temperature dependent surface heat flux. Chiu and Chen [10] used a decomposition method for solving the convective longitudinal fins with variable thermal conductivity. ADM also have used by several researchers to solve a wide range of physical problems in various engineering fields such as fluid flow and porous media simulation [6,12,14,17].

The Principle of the Adomian Decomposition Method (ADM) [3]
Beginning with an equation where F represents a general nonlinear ordinary differential operator involving both linear and nonlinear terms. The linear term is decomposed into L + R, where L is easily invertible and R is the remainder of the linear operator. For convenience, L may be taken as the highest order derivative which avoids difficult integrations which result when complicated Green's functions are involved. Thus the equation (1) can be written as Lu+Ru+Nu=g (2) where Nu represents the nonlinear terms. Solving for Lu, Because L is invertible, operating with its inverse L -1 yields An equivalent expression is where Φ is the integration constant and satisfies LΦ=0. For initial-value problems we conveniently for as the n-fold definite integration operator from t0 to t.
For the operator for example, we have For boundary value problems, indefinite integrations are used and the constants are evaluated from the given conditions. Solving for u yields The Adomian decomposition method [22] assumes an infinite series solution for unknown function u given by  (8) and the nonlinear term Nu, assumed to be analytic function f(u), is decomposed as follows: where An are the appropriate Adomian's polynomials. These An polynomials depend on the particular nonlinearity and these An Adomian polynomials are calculated by the general formula Substituting eq. (8) and eq. (9) into eq. (5) gives Each term of series (8) is given by the recurrence relation where An are the special Adomian polynomials or equivalently . . . (13) So, the practical solution for the n-term approximation is, (14) and the exact solution is

The Adomian decomposition method applied to Newell-Whitehead model
The nonlinear wave equation with dissipation and nonlinear transport term is given as: [15], [16] with the initial and boundary conditions = is nonlinear term. Applying the inverse operator to the equation (19) and using the initial data (18a) yields The ADM suggests the solution ) , ( t The first four components of Adomain polynomials according to (13) read (23) From eq. (21) and eq. (22), the iterates are determined by the following recursive way: The decomposition method provides a reliable technique that requires less work if compared with traditional techniques.

Application and Numerical Results
To give a clear overview of the methodology, the following example will be discussed. All the results are calculated by using the MATLAB 7.4 software. Consider the Newell-Whitehead equation [11]

Conclusions
The Adomain decomposition method is effective and powerful method for solving nonlinear partial differential Newell-Whitehead equations. The important part of this method is calculating Adomain polynomials for nonlinear operator.