USING ADOMIAN DECOMPOSITION METHOD FOR SOLVING SYSTEMS OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we introduce application of Adomian Decomposition Method (ADM) for solving systems of Ordinary Differential Equations (ODEs). This method is illustrated by four examples of (ODEs) and solutions are obtained. One of the most important advantages of this method is its simplicity in using.


INTRODUCTION
The literature on the Adomian decomposition method (ADM) and its modifications [1][2][3][4][5][6][7] tells us that this method is proven to be efficient to solve linear and nonlinear ODEs, DAEs, PDEs, SDEs, integral equations and integrointegral equations. More importantly, such method has been applied to a wide class of problems in physics, biology and chemical reaction. The reason of such spread and application of the method lies in the fact that the ADM provides the solution in a rapid convergent series with computable terms. In this manuscript, we aim at introducing a new reliable modification of ADM. For this reason, a new differential operator for solving high-order and system of differential equations. In order to illustrate the application of the modified form of the ADM, we would provide a set of examples to show the advantages of using the proposed method to solve the initial value problems.

ANALYSIS OF THE ADM
We consider the following system of ordinary differential equations of second order . . .
With the following initial conditions . . .
where f 1 , f 2 , ... f i are nonlinear functions, p i (x),and g i (x) are given functions.
According to the ADM we rewrite the system of equations(1) in terms of operator from as . . .
where L i are differential operators given by and their inverse integral operators are defined as Applying L −1 i on (2) we get . . .
where A in are the Adomian polynomials [8] are given
then we define: From (6) and (8), we can determine the components u n , v n , w n , ... can be immediately obtained.

APPLICATIONS OF THE METHOD
In this section, we will provide four numerical examples that shows this method.
Consider the system of linear second order ordinary differential equations: with initial conditions The exact solution is In an operator form eq.(9) became Applying L −1 on both side of eq.(10) and using the initial conditions, we get We use the following scheme Therefore Approximations to the solution of the above system with three iterations of ADM, yields: In this example, we note the solution by ADM close to the exact solution.

Example 2.
We study the system of nonlinear equation of Emden-Fowler type with initial conditions with the exact solution see in [9] ( where p 1 (x) = 1 x ,p 2 (x) = 2 x we find the inverse operators L −1 are given by applying the inverse operators L 1 ,L 2 on (11) and using the initial conditions we get We use the following scheme where A 1n , A 2n are Adomian polynomials that represent nonlinear term. Which are given by The comonents of the Adomian polynomials are given by ... and the nonlinear term v 2 ,has the few Adomian polynomials A 2n are given by Approximations to the solutions are as follows:

leads to
From the previous example we note that, the solution by ADM converges to the exact solution.
We sutdy the system of nonlinear equations of Emden-Fowler type with initial conditions The exact solutions see in [9] are x . System (14) we can write as where Lu, Lv define by: And the inverse operators L −1 define by: Applying L −1 on equation (15), and using the initial conditions, we get by assuming that
as well as and Therefore This gives the exact solution of Eq.(14) which is given as follows Example 4.
Consider the system of non-liner equations: with initial conditions The exact solutions are (u(x), v(x)) = (e −x , e x ).