INTRODUCTION
In this study, we consider the Abelian differential equations of the type:
with initial condition
where, f_{j} (x), j = 0,..., 3, are given continuous, linear or nonlinear functions and c is a real finite constants.
The Adomian Decomposition Method (ADM) (Adomian, 1988, 1991) will be effectively
used to approach Eq. 1. The Adomian algorithm introduces
the solution in the form of a rapidly convergent series with elegantly computable
terms. In the literature, the decomposition method has been used to obtain approximate
solutions of a large class of linear or nonlinear differential equations (Adomian,
1988, 1991, 1994; Adomian et al., 1991; Adomian, 1994). This computational
method yields analytical solutions and has certain advantages over standard
numerical methods. It is free from rounding off errors as it does not involve
discretization and does not require large computer obtained memory or power.
ADM consists of splitting the given equation into linear and nonlinear parts, inverting the highestorder derivative operator contained in the linear operator on both sides, identifying the initial and\or boundary conditions and the terms involving the independent variable alone as initial approximation, decomposing the unknown function into a series whose components are to be determined, decomposing the nonlinear function in terms of special polynomials called Adomian`s polynomials and finding the successive terms of the series solution by recurrent relation using Adomian`s polynomials. ADM is quantitative rather than qualitative, analytic, requiring neither linearization nor perturbation and continuous with no resort to discretization and consequent computerintensive calculations. The method is well addressed in (Guellal and Cherruault, 1995).
The convergence of the decomposition series has been investigated by several
researchers, Abbaoui and Cherruault (1996), Adomian (1988), Seng et al.
(1996), Cherruault (1989) and Cherruault and Adomain (1993). They obtained some
results about the speed of convergence of this method. Abbaoui and Cherruault
(1996) have proposed a new approach of convergence of the decomposition series.
The authors have given a new condition for obtaining convergence of the decomposition
series to the classical presentation of the ADM in Abbaoui and Cherruault (1996).
In this study we use ADM for obtaining approximate solutions of the Abelian
differential equation. In the case of f_{3} (c) = 0, Eq.
1 reduces to the well known Riccati equation. The importance of this equation
usually arises in the optimal control problems. The feed back gain of the linear
quadratic optimal control depends on a solution of a Riccati differential equation
which has to be found for the whole time horizon of the control process (Zill,
1997; Bulut and Evans, 2002) applied the decomposition method for solving Riccati
differential equation. ElTawil et al. (2004) applied the multistage
Adomian,s decomposition method for solving Riccati differential equation and
compared the results with standard ADM. Recently, Momani and coworkers extended
the method to fractional differential equations, (Momani, 2006; Momani and Shawagfeh,
2006; Momani and Qaralleh, 2007). In this study, the Adomian Decomposition Method
(ADM) proposed in (Adomian, 1988, 1991) is used to handle the nonlinear differential
Equations (1). Comparisons are also made between the present method and fourthorder
RungeKutta method to show the efficiency of the present method.
ANALYSIS OF METHOD
Here, we review the Adomian decomposition method by following the procedures
of (Adomian, 1988, 1991). However the same techniques can be applied on other
system of equations (Bulut and Evans, 2002). To introduce this method, we consider
the operator equation Fu = G, where F represents a general nonlinear ordinary
differential operator and G is a given function. The linear part of F can be
decomposed into L + R, where L is easily invertible and R is the remainder of
F. Thus the equation may be written as:
where, N is a nonlinear operator, L is the highestorder derivative which is assumed to be invertible, R is a linear differential operator of the order less than L and G is the source term.
The method is based on applying the operator L^{1} formally to the
expression
So by using the given conditions we obtain
where, h is the solution of the homogeneous equation Lu = 0, with the initial/boundary conditions.
The problem now is the decomposition of the nonlinear term Nu. To do this, Adomian developed a very elegant technique as follows:
Define the decomposition parameter λ as , then N (u) will be a function
of λ, u_{0}, u_{1},..., next expanding N (u) in Maclurian
series with respect to λ we obtain , where
where, the components of A_{n} are the socalled Adomian polynomials,
they are generated for each nonlinearity, for example, for N (u) = f(u), the
Adomian polynomials are given as:
They can be found from the formula (Adomian, 1990):
where, f^{(k)} (u_{0}) = d^{k}f(u_{0})/ d_{u}^{k} and
C (k, n) means the sum of possible products of k components of u, whose subscripts
add to n, divided by the factorial of the number of repetitions.
Now, we parameterize Eq. 3 in the form
where, λ is just an identifier for collecting terms in a suitable way
such that u_{n} depends on u_{0}, u_{1},..., u_{n1}
and we will later set λ = 1
Equating the coefficients of equal powers of λ, we obtain
And in general
Finally an Nterm approximate solution is given by
and the exact solution is .
APPLICATION AND NUMERICAL RESULTS
The decomposition method requires that the Abelian differential Eq.
1 be expressed in terms of operator form as
where, .
The ADM suggests the solution y(x) be decomposed by the infinite series of
components
and the nonlinear terms
where, A_{n} and B_{n} are the Adomain polynomials of y^{2},
y^{3}, respectively. The first four components of Adomain polynomials
reads
Substitution the decomposition series Eq. 13 and 14
into both sides of (12), then applying the inverse L^{1} gives:
From this equation, 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. To give a clear overview of the methodology,
the following two examples will be discussed. All the results are calculated
by using the symbolic calculus software Mathematica.
Example 1: Consider the following Abelian differential equation
subject to the initial condition
Substituting Eq. 18 and the initial condition Eq.
19 into Eq. 17 and using Eq. 16
to calculate the Adomian polynomials, yields the following recursive relation
Using the above recursive relationship and Mathematica, the first few terms of the decomposition series are given by:
other components are determined similarly. The fourthterm approximate solution
is given by
The graphs of the approximate solution Eq. 21 and the solution
obtained by fourthorder RungeKutta method are given in Fig.
1. It can be seen that the results from the ADM match the results of the
RungeKutta method very well, which implies that the ADM can predict the motion
of the Abelian differential equation accurately for the region under consideration.
Example 2: Consider the following Abelian differential equation

Fig. 1: 
Plots of y versus x. RungeKutta method (—); Eq.
21 (  ) 
subject to the initial condition
Substituting Eq. 22 and the initial condition Eq.
23 into Eq. 17 and using Eq. 16
to calculate the Adomian polynomials, yields the following recursive relation
Using the above recursive relationship and Mathematica, the first few terms of the decomposition series are given by
other components are determined similarly. The fifthterm approximate solution
is given by

Fig. 2: 
Plots of y versus time x. RungeKutta method (—); Eq.
(25) (  ). 
Numerical results are shown in Fig. 2. It is clear, as in the previous case, that the ADM solution works very well.
Errors are small and may be made smaller by using more terms of the ADM truncated
series.
CONCLUSIONS
In this study, the Adomian decomposition method has been successfully applied
to finding the approximate solution of the Abelian differential Eq.
1. All of the examples show that the results of the present method are in
excellent agreement with those obtained by the fourth order RungeKutta method.
The Adomian decomposition method was clearly very efficient and powerful technique
in finding the solutions of the proposed equation. The reliability of the method
and reduction in the size of computational domain give this method a wider applicability.