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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 H. Akca, E. A. Al-Zahrani and 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.

MSC:
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
References:
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[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)
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[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)