This paper will consider the implementation of fifthorder direct method in the form of AdamsMoulton method for solving directly secondorder delay differential equations (DDEs). The proposed direct method approximates the solutions using constant step size. The delay differential equations will be treated in their original forms without being reduced to systems of firstorder ordinary differential equations (ODEs). Numerical results are presented to show that the proposed direct method is suitable for solving secondorder delay differential equations.
In the recent years, there are rigorous and numerous researches undertaken in the areas of science and engineering that are skewed towards the developments of the mathematical models involving the delay differential equations. In mathematics, the DDEs are differing from ODEs in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. Several numerical methods have been proposed to solve firstorder DDEs such as in [
In this paper, we are concerned with solving secondorder delay differential equations (DDEs) as follows:
The direct AdamsMoulton methods were studied by several researchers and the methods have shown their ability to solve first, second and higherorder ODEs [
Most numerical methods for ODEs can be used to solve DDEs. In this paper, we adopted direct method proposed by Majid and Suleiman [
The point
Integrating once
Integrating twice
The function
Predictor
Corrector
The order of this developed method is calculated by referring to [
The formulae is defined as
The method has order
By referring to the corrector formulae in (
Hence,
The corrector of the direct method is of order five and the error constant is
The method is zero stable provided the roots
We rewrite (
The first characteristic polynomial of the method is given as
Since
There are many concepts of stability for numerical methods when applied to DDEs, depending on the test equation as well as the delay term involved. We would like to study the stability of the method by substituting the following test equation:
We solve the determination of
The following stability polynomial is obtained by letting
The boundary of the stability region in
Stability region of the direct method.
A basic property for an effective numerical method is that the method needs to converge. A linear multistep method is convergent if and only if it is stable and consistent [
By definition, a linear multistep method of the form
The proposed direct method is of order five, where
Since the method is order five which is ≥1, therefore, the method is consistent according to the definition. In the previous section, it has been shown that the method is zero stable. Therefore, we can conclude that this method is convergent.
In the code of PECE scheme
In the code, the selection of step sizes is predetermined.
Set starting value
For
Evaluate the approximate value
While
Set
Evaluate the approximate value
Computing
Computing
Complete.
In order to study the efficiency of the proposed direct method, we presented three secondorder DDE problems with constant delay in the following test problems. The numerical results of the direct method when solving Problems
Consider
Exact solution
Consider
Exact solution
Consider
Exact solution
The algorithm of the C language was executed on the Microsoft Visual C++ environment. The notations at the end of the paper are used in Tables
Comparison of the numerical results for solving Problem

Cubic spline [ 
Direct method 


MAXE  MAXE  FCN  
0 


122 
1 


288 
2 


494 
3 


819 
4 


1231 
5 


1960 
6 


3331 
Comparison of the numerical results for solving Problem
Variable multistep method [ 
Direct method  


MAXE 


MAXE  FCN 
0.1280 

0.1280  0.01280 

61 
0.2360 

0.2360  0.02360 

67 
0.3418 

0.3418  0.03418 

67 
0.4444 

0.4444  0.04444 

67 
0.5440 

0.5440  0.05440 

67 
0.6412 

0.6412  0.06412 

67 
0.7346 

0.7346  0.07346 

67 
0.8264 

0.8264  0.08264 

69 
0.9136 

0.9136  0.09136 

73 
1.0000 

1.0000  0.10000 

73 
Comparison of the numerical results for solving Problem
MATLAB solver dde23  Direct method  

RT  AT  MAXE  TS  TFS  FCN 

MAXE  TS  TFS  FCN 



20  1  64  Pi/10 

7  0  62 



37  1  115  Pi/20 

17  0  82 



74  0  233  Pi/50 

47  0  142 



158  0  475  Pi/100 

97  0  242 



340  0  1021  Pi/200 

197  0  442 



732  0  2197  Pi/250 

247  0  542 



1577  0  4732  Pi/500 

497  0  1042 
The numerical results for solving Problems
In Problem
Comparison of maximum error between cubic spline and direct method for solving Problem
Table
Comparison of maximum error between variable multistep method and direct method for solving Problem
Table
Comparison of maximum error and function calls between dde23 and direct method for solving Problem
In this study, we have shown that the proposed direct AdamsMoulton method using constant step size is suitable for solving secondorder DDEs directly. The proposed direct method has solved the secondorder DDEs in their original forms without being reduced to firstorder ODEs. This approach has given advantage in terms of computational cost to the direct method. The method has shown superiority in terms of accuracy and it has less computational cost.
Step size
Magnitude of the maximum absolute error
Total function calls
Total steps
Total failure steps
Real tolerance
Absolute tolerance
MATLAB solver dde23 based on the explicit RungeKutta (2,3) pair [
The authors gratefully acknowledged the financial support of University Putra Malaysia Grant.