Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method

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.


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-order 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 secondorder random differential equation.Section 4 is devoted to numerical examples.

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, ( ) ( ) where x t is an unknown function, ( ),  is a nonlinear operator, ( ) g t is the source in homogeneos term, and 0 , A y and 1 y are random variables.We now write the differential transform of function as In fact, ( ) x t is a differential inverse transform of the form It is clear from (3) and (4) that the concept of differential transform is derived from Taylor series expansion.That is ( ) ( ) Original function Transformed function ( ) ( ) 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 ( ) U k , respectively, are the transformed functions of ( ) x t , ( ) y t and ( ) u t .The fundamental mathematical operations of differential transformation are listed in the following table.

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 P L 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 ( ) , , is the set of second order random variables.Then the random variable , where [ ] E  is an expectation operator.The norm on is denoted by 2  .For example, for the random variable X we define ( ) , L X is a Banach space.In addition, let T (real interval) represent the space of times, we say that A sequence of second order random variables { } 4), we truncate the expansion of at the term as follows 0 ( ) ( ) By using the independence between 0 , A y and 1 y we have where The following Lemma guarantee the convergent of the sequence ( )  and the sequence

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 independently of the initial conditions 0 Y and 1 and [ ] The approximate mean and variance are ( ) ( ) where A is a Beta r.v. with parameters      ( ) ( ) where A is a uniform r.v. with parameters and independently of the initial conditions o Y and 1 Y which are independent r.v.'s satisfy [ ]      ( )

Conclusion
In this paper, we successfully applied the differential transform method to solve the second-order random    1)-( 2) with coefficients which depend on a random variable A which has been assumed to be independent of the random initial conditions 0 y and 1 y .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.

1 Y
are independent r.v.'s such as [ ]

Figure 3 Example 3 : 1 Y
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

Figure 1 . 1 Figure 2 .
Figure 1.Graphs of the expectation approximation solution of the DTM with n = 18.

Figure 3 .
Figure 3. Graphs of the expectation approximation solution of the DTM with n = 18.

Figure 4 .
Figure 4. Graphs of variance approximation solution of the DTM with n = 18.

Figure 5 Example 4 :and 1 Y
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

Figure 7 Example 5 :
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

Figure 9
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 .
Figure 5. Graphs of the expectation approximation solution of the HAM with n = 18.

Figure 8 . 1 Y 1 oFigure 9 .
Figure 8. Graphs of variance approximation solution of the DTM with n = 18.initial conditions o Y and 1 Y which satisfy [ ] 1 o E Y = ,

Figure 10 .
Figure 10.Graphs of variance approximation solution of the DTM with n = 18.

Figure 11
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.

Figure 11 .
Figure 11.Graphs of the expectation approximation solution of the DTM with n = 18.

Figure 12 .
Figure 12.Graphs of variance approximation solution of the DTM with n = 18.
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.
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