Journal of Applied Mathematics and Physics
Vol.07 No.03(2019), Article ID:91102,9 pages
10.4236/jamp.2019.73038
Solving the Linear Oscillatory Problem without Damping with Random Loading Condition Using the Decomposition Method
Amnah S. Al-Juhani*, Aleh A. Al-Shammari
Faculty of Science, Tabuk University, Tabuk, KSA
Copyright © 2019 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: April 2, 2018; Accepted: March 10, 2019; Published: March 13, 2019
ABSTRACT
In this paper we study the solution of random linear oscillatory equation without damping and with random leading condition using the method. Finally, the time evolution of the mean, variance and standard deviation has been plotted for a range of values of the natural frequency w.
Keywords:
Linear Stochastic Differential Equations, Adomian Decompositions, Linear Oscillatory, Mathematica
1. Introduction
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) [1] - [12] . 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 [13] - [20] . 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 [21] [22] [23] [24] [25] .
2. Problem Formulation
In this paper, we focus on solving the following Solving the linear oscillatory problem
(1)
(2)
under stochastic excitation with the deterministic initial conditions
where
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:
(3)
(4)
By obtaining the P.d.f. of , the average and variance of the solution process in terms of t: time, the general solution is
(5)
The ensemble average is given by
(6)
The covariance takes the form
(7)
The variance is
(8)
Due to linearity and the deterministic properties of and the frequency w we obtain a Gaussian solution process:
(9)
where .
Equation (9) represents a closed form solution of problem (1) with random loading condition.
3. The Adomian Decomposition Method
Case-study:
Let us consider
(10)
In the Adomian decomposition method, differential operators are decomposed. Thus Equation (1) is rewritten in the following form:
(11)
where:
Hence,
(12)
Solving for x we obtain
(13)
where is the solution of
(14)
Subject to the initial conditions:
(15)
Thus, the solution of equation takes the form:
(16)
We now assume that the solution can be written in the following form:
(17)
Substituting (17) in (16) we obtain:
(18)
By matching the boundaries, we obtain:
(19)
(20)
(21)
And the nth term will be:
(22)
By applying this procedure to equation, we obtain:
(23)
(24)
(25)
(26)
The nth term is:
(27)
Thus,
(28)
where,
(29)
(30)
(31)
(32)
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
(33)
Example:
Let us consider
(34)
in the previous case-study. By using the decomposition method, the following results are obtained (Figures 1-8).
Cite this paper
Al-Juhani, A.S. and Al-Shammari, A.A. (2019) Solving the Linear Oscillatory Problem without Damping with Random Loading Condition Using the Decomposition Method. Journal of Applied Mathematics and Physics, 7, 527-535. https://doi.org/10.4236/jamp.2019.73038
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