IJPAM: Volume 112, No. 1 (2017)

Title

ON THE STABILITY OF THE SOLUTIONS FOR A DELAY
DIFFERENTIAL EQUATIONS WITH DISCONTINUITY

Authors

A. Ahmad$^1$, N. Javaid$^2$, M. Rafaqat$^3$, A. Zeinev$^4$
$^{1,2,3}$Abdus Salam School of Mathematical Sciences (ASSMS)
GC University
68-B, New Muslim Town, Lahore, IR PAKISTAN
$^4$Department of Mathematics
University of Chemical Technology and Metallurgy (UCTM)
8, St. Kl. Ohridski, Blvd., 1756 Sofia, BULGARIA

Abstract

A numerical method of solving delay differential equations with fixed time delay and variable discontinuities of the solutions is considered. Runge-Kutta methods of higher order are used. The effectiveness of the method is shown by an example with proper initial data. The existence of a stable solution is discussed.

History

Received: September 11, 2016
Revised: October 27, 2016
Published: January 26, 2017

AMS Classification, Key Words

AMS Subject Classification: 34A37, 34K45
Key Words and Phrases: functional differential equations, impulsive differential equations, Runge-Kutta methods, delay, stability

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Bibliography

1
V. Acary, B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics, Springer-Verlag, Berlin-Heidelberg, 2008.

2
R. Baier, Q. Din, T. Donchev, Higher order Runge-Kutta methods for impulsive differential systems, Applied Mathematic and Computations, 218 (2012), 11790-11798.

3
R. Baier, T. Donchev, Discrete approximations of impulsive differential inclusions, Num. Funct. Anal. Optim., 31 (2010), 653-678.

4
D. Bainov, S. Hristova, Differential equations with MAXIMA, Francis and Taylor, 2011.

5
M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Company, New York, 2006.

6
A. Bellen, M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford Science Publications, Oxford, 2003.

7
B. Balachander, T. Nagy, D. Gilsinn, Delay differential equations, Springer, 2009.

8
J. Butcher, Numerical Methods of Ordinary Differential Equations, Wiley, West Sussex, 2003.

9
Q. Din, T. Donchev, D. Kolev, Numerical approximations of impulsive delay differential equations, Numerical Functional Analysis and Optimization, 34, No. 7 (2012), 728-740.

10
A. Dishliev, K. Dishlieva, S. Nenov, Specific asymptotic properties of the solutions of impulsive differential equations. Methods and applications, Academic Publication, 2012.

11
T. Donchev, N. Kitanov, D. Kolev, Stability for the solutions of parabolic equations with MAXIMA, PanAmerican Mathematical Journal, 20, No. 2 (2010), 119.

12
T. Donchev, A. Nosheen, Fuzzy functional differential equations of population dynamics, Ukr. Math. Journal 65 (2013), 787-79.

13
T. Donchev, D. Kolev, A. Nosheen, M. Rafaqat, A. Zeinev, Numerical Methods for Delayed Differential Equations with Discontinuities, Pliska Studia Mathematica Bulgarica (2014), 55-65.

14
J. Hale, Theory of functional differential equations, Springer-Verlag, 1977.

15
F. Ismail, R. Al-Khasawneh, A. Lwin, M. Suleyman, Numerical treatment of delay differential equations by Runge-Kutta method using Hermite interpolation, Matematika, 18 (2002), 79-90.

16
W. Schirotzek, Nonsmooth Analysis, Springer-Verlag, Berlin-Heidelberg, 2007.

17
T. Yang, Impulsive control theory, Springer-Verlag Berlin-Heidelberg, 2001.

How to Cite?

DOI: 10.12732/ijpam.v112i1.16 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 112
Issue: 1
Pages: 205 - 218


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