# Solving Delay Differential Equations Using Reformulated Block Backward Differentiation Formulae Methods

## Abstract

In this paper, the conventional backward differentiation formulae methods for step numbers k = 3 and 4 were reformulated by shifting them one-step backward to produce two and three approximate solutions respectively, in a step when implemented in block form. The derivation of the continuous formulations of the reformulated methods were carried out through multistep collocation method by matrix inversion technique. The discrete schemes were deduced from their respective continuous formulations. The convergence analysis of the discrete schemes were discussed. The stability analysis of these schemes were ascertained and the P- and Q-stability were also investigated. When the discrete schemes were implemented in block form to solve some first order delay differential equations together with an accurate and efficient formula for the solution of the delay argument, it was observed that the results obtained from the schemes for step number k = 4 performed slightly better than the schemes for step number k = 3 when compared with the exact solutions. More so, on comparing these methods with some existing ones, it was observed that the methods derived performed better in terms of accuracy.

Keywords:
Delay differential equations, reformulated block method, backward differentiation formulae, continuous formulations

## Article Details

How to Cite
Sirisena, U., & Yakubu, S. (2019). Solving Delay Differential Equations Using Reformulated Block Backward Differentiation Formulae Methods. Journal of Advances in Mathematics and Computer Science, 32(2), 1-15. https://doi.org/10.9734/jamcs/2019/v32i230139
Section
Original Research Article

## References

Al-Mutib AN. Numerical methods for solving delay differential equations. Ph.D. Thesis. University of Manchester, United Kingdom; 1977.

Oberle HJ, Pesh HJ. Numerical treatment of delay differential equations by Hermite interpolation. Numer. Math. 1981;37:235–255.

Thompson S. Step size control for delay differential equations using continuously imbedded Runge-Kutta methods of Sarafyan. Journal of Computational and Applied Mathematics. 1990;31:267-275.

Suleiman MB, Ismail F. Solving delay differential equations using component wise partitioning by Runge- Kutta method. Applied Mathematics and Computation. 2001;122:301-323.

Ishak F, Suleiman MB, Omar Z. Two-point predictor-corrector block method for solving delay differential equations. Matematika. 2008;24(2):131-140.

Ishak F, Majid ZA, Suleiman M. Two-point block method in variable step size technique for solving delay differential equations. Journal of Materials Sc. and Eng. 2010;4(12):86–90.

Radzi HM, Majid ZA, Ismail F, Suleiman M. Two and three point one-step block methods for solving delay differential equations. Journal of Quality Measurement and Analysis. 2012;82(1):29–41.

Bellen A, Zennaro M. Numerical methods for delay differential equations. New York: Oxford University Press; 2003.

Onumanyi P, Awoyemi DO, Jator SN, Sirisena UW. New linear multistep methods with continuous coefficients for first order initial value problems. Journal of Nigerian Mathematical Society. 1994;13: 37-51.

Karline S, Jeff C, Francesca M. Solving differential equations in R. Springer-Verlag Berlin Heidelberg; 2012.

Heng SC, Ibrahim ZB, Suleiman MB, Ismail F. Solving delay differential equations by using implicit 2-Point Block Backward Differentiation Formulae. Pentica J. Sci & Technol. 2013;21(1):37-44.