The Adomian decomposition technique was firstly introduced by Adomian in 1975. This technique can be used to solve differential, integral, algebraic and many other equations (linear or nonlinear)  -  . The method is based on a suggestion by Adomian G. that the solution can be decomposed into components. In the coming sections we will see that the Adomian decomposition method is also very convenient computationally and offers some significant advantages  -  . The Adomian decomposition method is not a perturbation procedure, so no assumption concerning the size of randomness is necessary, where each term from the decomposed solution depends only on the preceding terms. A little work in the convergence of the procedure had been done      .
2. Problem Formulation
In this paper, we focus on solving the following Solving the linear oscillatory problem
under stochastic excitation with the deterministic initial conditions
w: frequency of oscillation,
: deterministic nonlinearity scale,
: a triple probability space with as the sample space, where σ is a σ-algebra on event in and P is a probability measure, and is a white noise with the following properties:
By obtaining the P.d.f. of , the average and variance of the solution process in terms of t: time, the general solution is
The ensemble average is given by
The covariance takes the form
The variance is
Due to linearity and the deterministic properties of and the frequency w we obtain a Gaussian solution process:
Equation (9) represents a closed form solution of problem (1) with random loading condition.
3. The Adomian Decomposition Method
Let us consider
In the Adomian decomposition method, differential operators are decomposed. Thus Equation (1) is rewritten in the following form:
Solving for x we obtain
where is the solution of
Subject to the initial conditions:
Thus, the solution of equation takes the form:
We now assume that the solution can be written in the following form:
Substituting (17) in (16) we obtain:
By matching the boundaries, we obtain:
And the nth term will be:
By applying this procedure to equation, we obtain:
The nth term is:
Figure 1. The mean of at .
Figure 2. The variance of at .
Figure 3. The covariance of at .
Figure 4. The covariance of at .
Figure 5. The mean of at .
Figure 6. The variance of at .
Figure 7. The covariance of at .
Figure 8. The covariance of at
Let us consider
in the previous case-study. By using the decomposition method, the following results are obtained (Figures 1-8).
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