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.

DOI: 10.4236/ojapps.2016.64028                

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

Received 21 March 2016; accepted 25 April 2016; published 28 April 2016

1. Introduction

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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