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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.

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 [

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.

The differential transform method (DTM) has been used by Zhou [

where

source in homogeneos term, and

We now write the differential transform of function as

In fact,

It is clear from (3) and (4) that the concept of differential transform is derived from Taylor series expansion. That is

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

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

we deal with the triplet Probabilistic space

order random variables. Then the random variable

the space of times, we say that

A sequence of second order random variables

To proceed from (4), we truncate the expansion of at the term as follows

By using the independence between

where

The following Lemma guarantee the convergent of the sequence

Lemma [

and

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

where

The approximate mean and variance are

Example 2: Consider random initial value problem

The approximate mean and variance are

Example 3: Consider the problem

The approximate mean and variance are

Example 4: Consider the problem

uniform r.v. with parameters

The approximate mean and variance are

Example 5: Consider the problem

The approximate mean and variance are

Example 6: Consider the problem

A is a uniform r.v. with parameters

initial conditions

The approximate mean and variance are

ConclusionIn this paper, we successfully applied the differential transform method to solve the second-order random

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

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