zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Euler scheme and its convergence for impulsive delay differential equations. (English) Zbl 1202.65088
The paper focuses on the numerical solution of linear impulsive delay differential equations (IDDEs) using a fixed stepsize Euler scheme. The authors begin with a brief review of existing literature on the numerical solution of impulsive differential equations. They believe that this paper is the first one on numerical methods of IDDEs. They justify their decision to use a fixed stepsize scheme and acknowledge insight gained from work with impulsive logistic equations by {\it H. Akca, E. A. Al-Zahrani} and {\it V. Covachev} [Electron. J. Differ. Equ. 2005, Conf. 12, 1--8, electronic only (2005; Zbl 1084.39002)]. In section 2 they introduce their new Euler scheme for IDDEs, which involves taking partition nodes and using a fixed stepsize. They prove their scheme to have convergence order 1 in section 3 and present an illustrative example in section 4 to demonstrate convergence to the exact solution.

65L05Initial value problems for ODE (numerical methods)
65L20Stability and convergence of numerical methods for ODE
34K28Numerical approximation of solutions of functional-differential equations
34K06Linear functional-differential equations
Full Text: DOI
[1] Ballinger, G.; Liu, X. Z.: Existence and uniqueness results for impulsive delay differential equations, Dyn. contin. Discrete impuls. Syst. 5, 579-591 (1999) · Zbl 0955.34068
[2] Berezansky, L.; Braverman, E.: Exponential boundedness of solutions for impulsive delay differential equations, Appl. math. Lett. 9, 91-95 (1996) · Zbl 0864.34060 · doi:10.1016/0893-9659(96)00100-0
[3] Liu, B.; Liu, X. Z.; Teo, K.; Wang, Q.: Razumikhin-type theorem on exponential stability of impulsive delay systems, IMAJ. appl. Math. 71, 47-61 (2006) · Zbl 1128.34047 · doi:10.1093/imamat/hxh091
[4] Shen, J. H.; Yan, J. R.: Razumikhin-type stability theorems for impulsive function differential equations, Nonlinear anal. 33, 519-531 (1998) · Zbl 0933.34083 · doi:10.1016/S0362-546X(97)00565-8
[5] Wang, Q.; Liu, X. Z.: Exponential stability for impulsive delay differential equations by razumikhin method, J. math. Anal. appl. 309, 462-473 (2005) · Zbl 1084.34066 · doi:10.1016/j.jmaa.2004.09.016
[6] Li, Y. K.; Xing, W. Y.; Lu, L. H.: Existence and global exponential stability of periodic solution of a class of neural networks with impulses, Chaos solitons fractals 27, 437-445 (2006) · Zbl 1084.68103 · doi:10.1016/j.chaos.2005.04.021
[7] Liu, M. Z.; Liang, H.; Yang, Z. W.: Stability of Runge -- Kutta methods in the numerical solution of linear impulsive differential equations, Appl. math. Comput. 192, 346-357 (2007) · Zbl 1193.65121 · doi:10.1016/j.amc.2007.03.044
[8] Wu, S. J.: The Euler scheme for random impulsive differential equations, Appl. math. Comput. 191, 164-175 (2007) · Zbl 1193.65010 · doi:10.1016/j.amc.2007.02.073
[9] Covachev, V.; Akça, H.; Yeniçeriog&caron, F.; Lu: Difference approximations for impulsive differential equations, Appl. math. Comput. 121, 383-390 (2001) · Zbl 1024.65065
[10] Bainov, D. D.; Kamont, Z.; Minchev, E.: Difference methods for impulsive functional differential equations, Appl. numer. Math. 16, 401-416 (1995) · Zbl 0823.65085 · doi:10.1016/0168-9274(95)00006-G
[11] Akca, H.; Al-Zahrani, E.; Covachev, V.: Asymptotic behavior of discrete solutions to impulsive logistic equations, Electron. J. Differ. equ. Conf. 12, No. 1 -- 8 (2005)