Open Journal of Applied Sciences
Vol.06 No.04(2016), Article ID:66044,11 pages
10.4236/ojapps.2016.64028
Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method
Ayad R. Khudair, S. A. M. Haddad, Sanaa L. Khalaf
Department of Mathematics, Faculty of Science, University of Basrah, Basrah, Iraq
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 21 March 2016; accepted 25 April 2016; published 28 April 2016
ABSTRACT
The differential transformation method (DTM) is applied to solve the second-order random differential equations. Several examples are represented to demonstrate the effectiveness of the proposed method. The results show that DTM is an efficient and accurate technique for finding exact and approximate solutions.
Keywords:
Random Differential Equations, Stochastic Differential Equation, Differential Transformation Method
1. Introduction
The ordinary differential equations which contain random constant or random variables are well known topics which are called the random ordinary differential equations. The subject of second-order random differential equations is one of much current interests due to the great importance of many applications in engineering, biology and physical phenomena (see, e.g. Chil’es and Delfiner [1] , Cort’es et al. [2] , Soong [3] and references therein). Recently, several first-order random differential models have been solved by using the Mean Square Calculus [2] [4] - [11] . Variety scientific problems have been modeled by using the nonlinear second-or- der random differential equations. However, most of these equations cannot be solved analytically. Thus, accurate and efficient numerical techniques are needed. There are several semi-numerical techniques which have been considered to obtain exact and approximate solutions of linear and nonlinear differential equations, such as adomian decomposition method (ADM) [12] , variational iteration method (VIM) [13] and homotopy perturbation method (HPM) [14] . We observe that semi-numerical methods are very prevalent in the current literature, cf. [12] - [14] .
The object of this work is to describe how to implement the differential transformation method (DTM) for finding exact and approximate solutions of the second-order random differential equations. To this end, the second-order random differential equations and the concept of the differential transformation method are presented in Section 2. In Section 3, we consider the statistical functions of the mean square solution of the second- order random differential equation. Section 4 is devoted to numerical examples.
2. Differential Transform Method
The differential transform method (DTM) has been used by Zhou [15] . This method is effective to obtain exact and approximate solutions of linear and nonlinear differential equations. To describe the basic ideas of DTM, we consider the second order random differential equation,
(1)
(2)
where,
is an unknown function,
is a nonlinear operator,
is the
source in homogeneos term, and and
are random variables.
We now write the differential transform of function as
(3)
In fact, is a differential inverse transform of the form
(4)
It is clear from (3) and (4) that the concept of differential transform is derived from Taylor series expansion. That is
(5)
Differential transform for some functions.
Notes that, the derivatives in differential transform method does not evaluate symbolically.
In keeping with Equations (3) and (4), let,
and
, respectively, are the transformed functions of
,
and
. The fundamental mathematical operations of differential transformation are listed in the following table.
3. Statistical Functions of the Mean Square Solution
Before proceeding to find the computation of the main statistical functions of the mean square solution of Equations (1) and (2) we briefly clarify some concept, notation, and results belonging to the so-called calculus. The reader is referred to the books by Soong [3] , Loeve [16] , and Wong and Hajek [17] . Throughout the paper,
we deal with the triplet Probabilistic space. Thus, suppose
is the set of second
order random variables. Then the random variable, if
, where
is an expectation operator. The norm on is denoted by
. For example, for the random variable X we define
, in such way that
is a Banach space. In addition, let T (real interval) represent
the space of times, we say that is a second order stochastic process, if the random variable
for each
.
A sequence of second order random variables converges to
, if
To proceed from (4), we truncate the expansion of at the term as follows
(6)
By using the independence between and
we have
(7)
(8)
where
The following Lemma guarantee the convergent of the sequence to
and the sequence
to
if the sequence the
converges to
.
Lemma [5] : Let and
be two sequences of 2-r.vs X and Y, respectively, i.e.,
and
then
. If
, then
,
and.
4. Numerical Examples
In this section, we adopt several examples to illustrate the using of differential transform method for approximating the mean and the variance.
Example 1: Consider random initial value problem,
and
where and independently of the initial conditions
and
which satisfy
,
,
,
and
.
The approximate mean and variance are
Figure 1 explain the Graph of the expectation approximation solution by using DTM with n = 18, while Figure 2 explain the Graph of variance approximation solution by using DTM with n = 18.
Example 2: Consider random initial value problem,
and
where A is a Beta r.v. with parameters
and
, i.e.
and the initial conditions
and
are independent r.v.’s such as
,
,
,
.
The approximate mean and variance are
Figure 3 explain the Graph of the expectation approximation solution by using DTM with n = 18, while Figure 4 explain the Graph of variance approximation solution by using DTM with n = 18.
Example 3: Consider the problem,
and
whereA is a Beta r.v. with parameters
and
, i.e.
and independently of the initial conditions
and
which are independent r.v.’ satisfy
,
,
,
.
The approximate mean and variance are
Figure 1. Graphs of the expectation approximation solution of the DTM with n = 18.
Figure 2. Graphs of variance approximation solution of the DTM with n = 18.
Figure 3. Graphs of the expectation approximation solution of the DTM with n = 18.
Figure 4. Graphs of variance approximation solution of the DTM with n = 18.
Figure 5 explain the Graph of the expectation approximation solution by using DTM with n = 18, while Figure 6 explain the Graph of variance approximation solution by using DTM with n = 18.
Example 4: Consider the problem,
and
where A is a
uniform r.v. with parameters and
, i.e.
and independently of the initial conditions
and
which are independent r.v.’s satisfy
,
,
,
.
The approximate mean and variance are
Figure 7 explain the Graph of the expectation approximation solution by using DTM with n = 18, while Figure 8 explain the Graph of variance approximation solution by using DTM with n = 18.
Example 5: Consider the problem,
and
where A is auniform r.v. with parameters
and
, i.e.
and independently of the initial conditions
and
which are independent r.v.’s satisfy
,
,
,
.
The approximate mean and variance are
Figure 9 explain the Graph of the expectation approximation solution by using DTM with n = 18, while Figure 10 explain the Graph of variance approximation solution by using DTM with n = 18.
Figure 5. Graphs of the expectation approximation solution of the HAM with n = 18.
Figure 6. Graphs of variance approximation solution of the HAM with n = 18.
Example 6: Consider the problem,
and
where
A is a uniform r.v. with parameters and
, i.e.
and independently of the
Figure 7. Graphs of the expectation approximation solution of the DTM with n = 18.
Figure 8. Graphs of variance approximation solution of the DTM with n = 18.
initial conditions and
which satisfy
,
,
,
and
.
The approximate mean and variance are
Figure 9. Graphs of the expectation approximation solution of the DTM with n = 18.
Figure 10. Graphs of variance approximation solution of the DTM with n = 18.
Figure 11 explain the Graph of the expectation approximation solution by using DTM with n = 18, while Figure 12 explain the Graph of variance approximation solution by using DTM with n = 18.5.
ConclusionIn this paper, we successfully applied the differential transform method to solve the second-order random
Figure 11. Graphs of the expectation approximation solution of the DTM with n = 18.
Figure 12. Graphs of variance approximation solution of the DTM with n = 18.
differential Equations (1)-(2) with coefficients which depend on a random variable A which has been assumed to be independent of the random initial conditions and
. This includes the computation of approximations of the mean and variance functions to the random solution. These approximations not only agree but also improve those provided by the Adomian Decomposition Method [12] , Variational Iteration Method [13] and Homotopy Perturbation Method [14] as we have illustrated through different examples. Otherwise, the differential transform method is very effective and powerful tools for the second-order random differential equation because it is a direct way without using linearization, perturbation or restrictive assumptions.
Cite this paper
Ayad R. Khudair,S. A. M. Haddad,Sanaa L. Khalaf, (2016) Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method. Open Journal of Applied Sciences,06,287-297. doi: 10.4236/ojapps.2016.64028
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