A quasilinearization technique for the solution of singularly perturbed delay differential equation

Volume 2, Issue 1, pp 1--7

Publication Date: 2017-12-21

http://dx.doi.org/10.22436/mns.02.01.01

Authors

Fevzi Erdogan - Department of Mathematics, Faculty of Sciences, Yuzuncuyil University, 65080, Van, Turkey
Mehmet Giyas Sakar - Department of Mathematics, Faculty of Sciences, Yuzuncuyil University, 65080, Van, Turkey

Abstract

This study deals with the singularly perturbed initial value problems for a quasilinear first-order delay differential equation. A quasilinearization technique for the appropriate delay differential problem theoretically and experimentally analyzed. The parameter uniform convergence is confirmed by numerical computations.

Keywords

Delay differential equation, singular perturbation, finite difference scheme, piecewise-uniform mesh, quasilinearization technique

References

[1] G. M. Amiraliyev, H. Duru, A note on a parameterized singular perturbation problem, J. Comput. Appl. Math., 182 (2005), 233–242.
[2] G. M. Amiraliyev, F. Erdogan, Uniform numerical method for singularly perturbed delay differential equations, Comput. Math. Appl., 53 (2007), 1251–1259.
[3] G. M. Amiraliyev, F. Erdogan, A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations, Numer. Algorithms, 52 (2009), 663–675.
[4] I. G. Amiraliyeva, F. Erdogan, G. M. Amiraliyev, A uniform numerical method for dealing with a singularly perturbed delay initial value problem, Appl. Math. Lett., 23 (2010), 1221–1225.
[5] A. Bellen, M. Zennaro, Numerical methods for delay differential equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, (2003).
[6] R. Bellman, K. L. Cooke, Differential-difference equations, Academic Press, New York-London, (1963).
[7] S.-N. Chow, J. Mallet-Paret, Singularly perturbed delay-differential equations, Coupled nonlinear oscillators, Los Alamos, N. M., (1981), North-Holland Math. Stud., North-Holland, Amsterdam, 80 (1983), 7–12.
[8] R. D. Driver, Ordinary and delay differential equations, Applied Mathematical Sciences, Springer-Verlag, New York- Heidelberg, (1977).
[9] F. Erdogan, A parameter robust method for singularly perturbed delay differential equations, J. Inequal. Appl., 2010 (2010), 14 pages.
[10] F. Erdogan, G. M. Amiraliyev, Fitted finite difference method for singularly perturbed delay differential equations, Numer. Algorithms, 59 (2012), 131–145.
[11] Y. Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA, (1993).
[12] A. Longtin, J. G. Milton, Complex oscillations in the human pupil light reflex with ”mixed” and delayed feedback, Nonlinearity in biology and medicine, Los Alamos, NM, (1987), Math. Biosci., 90 (1988), 183–199.
[13] M. C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287–289.
[14] J. Mallet-Paret, R. D. Nussbaum, A differential-delay equation arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249–292.
[15] J. Mallet-Paret, R. D. Nussbaum, Multiple transition layers in a singularly perturbed differential-delay equation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 1119–1134.
[16] S. Maset, Numerical solution of retarded functional differential equations as abstract Cauchy problems, J. Comput. Appl. Math., 161 (2003), 259–282.
[17] B. J. McCartin, Exponential fitting of the delayed recruitmentrenewal equation, J. Comput. Appl. Math., 136 (2001), 343–356.
[18] H.-J. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl., 270 (2002), 143–149.

Downloads

XML export