# Numerical Solution and Stability of Block Method for Solving Functional Differential Equations

• Fuziyah Ishak
• Mohamed B. Suleiman
• Zanariah A. Majid
Conference paper

## Abstract

In this article, we describe the development of a two-point block method for solving functional differential equations. The block method, implemented in variable stepsize technique produces two approximations simultaneously using the same back values. The grid-point formulae for the variable steps are derived, calculated and stored at the start of the program for greater efficiency. The delay solutions for the unknown function and its derivative at earlier times are interpolated using the previous computed values. Stability regions for the block method are illustrated. Numerical results are given to demonstrate the accuracy and efficiency of the block method.

## Keywords

Block method Delay differential equation Functional differential equation Neutral delay differential equation Polynomial interpolation Stability region

## References

1. 1.
R.D. Driver, Ordinary and Delay Differential Equations (Springer, New York, 1977)
2. 2.
J.R. Ockendon, A.B. Taylor, The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. Lond., Ser. A 322, 447–468 (1971)Google Scholar
3. 3.
A. Iserles, On the generalized pantograph functional differential equation. Eur. J. Appl. Math. 4, 1–38 (1992)
4. 4.
F. Ishak, M.B. Suleiman, Z.A. Majid, Block method for solving pantograph-type functional differential equations, in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering 2013, WCE 2013, 3–5 July 2013, London, UK, pp. 948–952Google Scholar
5. 5.
D.J. Evans, K.R. Raslan, The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 82(1), 49–54 (2005)
6. 6.
W.S. Wang, S.F. Li, On the one-leg θ-methods for solving nonlinear neutral functional differential equations. Appl. Math. Comput. 193, 285–301 (2007)
7. 7.
I. Ali, H. Brunner, T. Tang, A spectral method pantograph-type delay differential equations and its convergence analysis. J. Comput. Math. 27(2–3), 254–265 (2009)
8. 8.
S. Sedaghat, Y. Ordokhani, M. Dehghan, Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Commun. Nonlinear Sci. Numer. Simul. 137, 4815–4830 (2012)
9. 9.
F. Samat, F. Ismail, M. Suleiman, Phase fitted and amplification fitted hybrid methods for solving second order ordinary differential equations. IAENG Int. J. Appl. Math. 43(3), 95–105 (2013)
10. 10.
C. Yang, J. Hou, Numerical method for solving volterra integral equations with a convolution kernel. IAENG Int. J. Appl. Math. 43(4), 185–189 (2013)
11. 11.
K.M. Hsiao, W.Y. Lin, F. Fuji, Free vibration analysis of rotating Euler beam by finite element method. Eng. Lett. 20(3), 253–258 (2012)Google Scholar
12. 12.
T.A. Anake, D.O. Awoyemi, A.O. Adesanya, One-step implicit hybrid block method for the direct solution of general second order ordinary differential equations. IAENG Int. J. Appl. Math. 42(4), 224–228 (2012)
13. 13.
Z.A. Majid, Parallel block methods for solving ordinary differential equations. Ph.D. thesis, Universiti Putra Malaysia (2004)Google Scholar
14. 14.
F. Ishak, M. Suleiman, Z. Omar, Two-point predictor-corrector block method for solving delay differential equations. Matematika 24(2), 131–140 (2008)Google Scholar
15. 15.
F. Ishak, Z.A. Majid, M. Suleiman, Two-point block method in variable stepsize technique for solving delay differential equations. J. Mater. Sci. Eng. 4(12), 86–90 (2010)Google Scholar
16. 16.
F. Ishak, Z.A. Majid, M.B. Suleiman, Development of implicit block method for solving delay differential equations, in Proceedings of the 14th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, Malta, 2012, pp. 67–71Google Scholar
17. 17.
C.T.H. Baker, C.A.H. Paul, Computing stability regions-Runge-Kutta methods for delay differential equations. IMA J. Numer. Anal. 14, 347–362 (1994)
18. 18.
A.N. Al-Mutib, Stability properties of numerical methods for solving delay differential equations. J. Comp. Appl. Math. 10, 71–79 (1984)
19. 19.
C.A.H. Paul, A test set of functional differential equations, in Numerical Analysis Report No. 243, February 1994Google Scholar

## Authors and Affiliations

• Fuziyah Ishak
• 1
• Mohamed B. Suleiman
• 2
• Zanariah A. Majid
• 3
1. 1.Faculty of Computer and Mathematical SciencesUniversiti Teknologi MARAShah AlamMalaysia
2. 2.Institute for Mathematical Research (INSPEM)Universiti Putra MalaysiaSerdangMalaysia
3. 3.Mathematics DepartmentUniversiti Putra MalaysiaSerdangMalaysia