INTRODUCTION

Retarded delay differential systems with proportional delays represent a particular class of delay differential systems. Such systems play an important role in the mathematical modeling of real world phenomena such as physical problems, circuit analysis, computer-aided design, power systems, simulation of mechanical systems and more general optimal control problems; thus, they have attracted the attention of numerical analysts (Hale and Lunel, 1993; Taiwo and Odetunde, 2010; Hafshejani et al., 2011; Shieh et al., 2011; Vanani et al., 2011a; Vanani and Aminataei, 2009, 2010). A RDDS is presented as follows (Bellen and Zennaro, 2003):

(1)

Where:

(2)

is the state vector and:

(3)

such that {α_k (x)≤b}^m_{k = 0} are delay functions; A(x) and B(x) are (m+1)-dimensional matrices which their entries are complex functions of x. Also:

(4)

(5)

represent the initial vector function and known vector function, respectively.

Obviously, most of these systems cannot be solved exactly. It is therefore necessary to design efficient numerical methods to approximate their solutions. TDM as a modification of ADM is considered as a efficient method for solving RDDSs. The ADM was first introduced by Adomian (1968, 1988) and has been used to integrate various systems of functional equations (Adomian, 1988). Recently, many literatures have been developed for the application of ADM (Adomian, 1988, 1994; Adomian and Rach, 1992; Chowdhury, 2011; Jaradat, 2008; Kooch and Abadyan, 2011, 2012). Several modifications of ADM have been presented various fields of applied mathematics and physics (Hosseini, 2006; Wazwaz, 1999a, b, 2000, 2002; Vanani et al., 2011b).

The difficult parts of ADM is to calculate the Adomian polynomials. There are large number of literature to present an efficient algorithm for computing Adomian polynomials (Wazwaz and El-Sayed, 2001). The most popular one is the formula obtained bu Adomian (1994, 1988) as:

(6)

where, A_n denotes the Adomian polynomial of degree n, is the exact solution of the problem and f(u) is the nonlinear term in the equation. It should b noted that the calculation of the Adomian polynomials is too difficult for large n and Eq. 6 can not be applied if f is a function of more than one variable, such as f = f(u, u') In addition, the ADM may diverge for some problems with special conditions (Hosseini and Nasabzadeh, 2006). Hence, to remove this difficulty we proposed a new and efficient method (TDM) for solving desired RDDS.

APPLICATION OF TDM ON RDDSs

The structure of TDM is as follows. Let us the problem (1) is given. So, we can consider its solution in the following form:

(7)

Where:

(8)

have to be determined sequentially upon the following algorithm:

(9)

and so on.

Adding the above equations, we obtain:

(10)

This method is called TDM which is useful for different problems in finite, infinite, regular and irregular domains. The convergency and more details of this method are given (Al-Refai et al., 2008).

ILLUSTRATIVE NUMERICAL EXPERIMENTS

Here, three experiments of RDDSs are given to illustrate the efficiency and validity of the method. All experiments are considered on the interval [0, 1]. To simplify the computations, we have used Taylor series expansion of each iteration. The computations associated with the experiments discussed below were performed in Maple 14 on a PC with a CPU of 2.4 GHz.

Experiment 1: Consider the RDDS (1) with the following conditions:

(11)

The exact solution is U(x) = [e^x, e^-x].

We have solved this problem using TDM with n = 4. The sequence of approximate solution is obtained as follows:

(12)

Hence, we get:

(13)

Therefore, we conclude that:

(14)

This has the closed form U(x) = [e^x, e^-x]^T which is the exact solution of the problem. To show the fastness of the method, the runtime of the proposed algorithm is also computed. Table 1 shows the results including the maximum absolute error and runtime of the method for different n.

Table 1 illustrates that the solutions of TDM are in good agreement with the exact solution. Also, the runtime of the proposed algorithm illustrate the method as a fast and powerful tool.

Experiment 2: Consider the RDDS (1) with the following conditions:

(15)

The exact solution is U(x) = [sin(x), e^x, cos(x)]^T.

We have solved this problem using TDM with n = 6. The sequence of approximate solution is obtained as follows:

Thus, we obtain:

(17)

Therefore, we conclude that:

(18)

This has the closed form U(x) = [sin(x), e^x, cos(x)]^T, which is the exact solution of the problem. The runtime of the method is also computed for different n to obtain a suitable approximation. Table 2 shows the results including the maximum absolute error and runtime of the method for different n.

The results also confirm the method as a fast method. Therefore, using the proposed method is preferred to facilitate the computations as shown in Table 2.

Table 1:	Maximum absolute error and runtime of the method for different n of experiment 1

Table 2:	Maximum absolute error and runtime of the method for different n of experiment 2

Experiment 3: Consider the RDDS (1) with the following conditions:

(19)

The exact solution is U(x) = [sinh(x), cosh(x), sin(x), e^x2]^T.

We have solved this problem using TDM with n = 7. The s following results are obtained:

(20)

Therefore, we conclude that:

(21)

This has the closed form U(x) = [sinh(x), cosh(x), sin(x) e^x2]^T which is the exact solution of the problem. Table 3 shows the results including the maximum absolute error and runtime of the method for different n.

The high accuracy of the method can be observed again. Therefore, we prefer the proposed method for solving RDDSs.

Table 3:	Maximum absolute error and runtime of the method for different n of experiment 3