International Journal of Physical Sciences

A numerical method based on the differential transform method (DTM) and the Adomian decomposition method (ADM) is introduced in this paper for the approximate solution of delay differential equation (DDE). The algorithm is illustrated by studying an initial value problem. The results obtained are presented and show that only few terms are required to obtain an approximate solution which is found to be accurate and efficient. 
 
   
 
 Key words: Delay differential equations, differential transform method, Adomian decomposition method.


The differential transform method (DTM)
The differential transformation of the k th -order derivative of function y(x) is defined as follows: (1) Where is the original function, is the transformed function and is the k th derivative with respect to x.The differential inverse transform of is defined as: (2) Combining Equations 1 and 2 we obtain (3) The following theorems that can be deduced from Equations 1 and 2 are given in Rostam et al. (2011).*Corresponding author.E-mail: kamal_raslan@yahoo.com.

Theorem 1
If the original function is , then the transformed function is .

Theorem 2
If the original function is , then the transformed function is .

Theorem 3
If the original function is , then the transformed function is .

Theorem 4
If the original function is then the transformed function is.

Theorem 5
If the original function is then the transformed function is .

Theorem 6
If the original function is then the transformed function is: .

Adomian decomposition method (ADM)
Consider the DDE of the form (El-Safty et al., 2003;Abdel-Halim and Ertürk, 2007) (4) (5) Where the differential operator is given by Raslan and Sheer 745 , (.) .)( The inverse operator 1  L is therefore considered as;  N fold integral operator defined by   remaining components of ) (x y can be determined in a way such that each component is determined by using the preceding components.In other words, the method introduces the recursive relation:

Theorem 7
The solution of the DDE in form of Equation 4 can be determined by the series of Equation 9with the iterations of Equation 10.To illustrate the Theorem 7, some examples of LDDE and NDDE are discussed in details and the obtained results are exactly the same with the one found by ADM.

APPLICATIONS
What follows examples for LDDEs and NDDEs will be examined by the two schemes presented above.Two physical models will be used for illustrative purposes regarding the comparison goal.

Linear delay differential equations
Example 1 Consider the first order LDDE in the form (Evans and Raslan, 2004) The first solution by DTM method Using Theorems 1, 2, 3, 4, 5 and 6, Equation 14transforms to   exact solutions for only a few terms.Comparing the DTM and the decomposition method with several other methods that have been advanced for solving DDEs, shows that the new technique is reliable, powerful and promising.We believe that the efficiency of the decomposition method gives it much wider applicability which needs to be explored further.


are constants describing the boundary.The ADM assumes that the unknown function ) (x y can be expressed as an infinite series of the form polynomials that can be generated for all forms of nonlinearity as follows: that arise from the boundary conditions at 0  x and from integrating the source term if it exists then secondly, the

For
above analysis yields the Theorem 7.
Comparisons have been made with known results as reported in

Table 1 .
The absolute errors in the approximation solutions using DTM and ADM (N = 13).

Table 1 .
It is clear fromTable 1 that the two methods not only give rapidly convergent series but also accurately compute the solutions.