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Numerical methods for impulsive differential equation. (English) Zbl 1145.65317

Summary: In this paper, the asymptotical stability of the numerical methods with the constant stepsize for impulsive differential equation

$\begin{array}{cc}\hfill \stackrel{˙}{x}\left(t\right)& =\alpha x,\phantom{\rule{2.em}{0ex}}t\ne k,t>0\hfill \\ \hfill {\Delta }x& =\sigma x,\phantom{\rule{2.em}{0ex}}t=k\hfill \\ \hfill x\left(0& +0\right)={x}_{0},\hfill \end{array}$

where $a\ne 0,\beta ,{x}_{0}\in ℝ,1+\beta \ne 0,k\in ℕ$, are investigated. The asymptotical stability conditions of the analytic solution of this equation and the numerical solutions are obtained. Finally, some experiments are given.

MSC:
 65L05 Initial value problems for ODE (numerical methods)
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