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An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations. (English) Zbl 1177.65197
The article deals with the Runge-Kutta method for the following neutral delay integro-differential equation $$\cases \frac{d}{dt}[y(t) - Ny(t - \tau)] = f\left(t,y(t),y(t - \tau)m{\int_{t-\tau}^t} g(t,s,y(s)) \, ds\right), \quad t \ge t_0 \\ y(t) = \varphi(t), \quad t_0 - \tau \le t \le t_0,\endcases$$ ($\varphi: \ [t_0 - \tau,t_0] \to {\Bbb C}^d$, $f: \ [t_0,\infty) \times {\Bbb C}^d \times {\Bbb C}^d \times \to {\Bbb C}^d$, $g: \ \{(t,s): \ t \in [t_0,\infty), s \in [t - \tau,t]\} \times {\Bbb C}^d \to {\Bbb C}^d$ are given functions). More precisely, the authors study the asymptotic stability of the Runge-Kutta method under small perturbations of the initial function $\varphi(t)$. The main result of the article desribes conditions under that the Runge-Kutta methods are globally and asymptotically stable. This result improves the analogous theorem by {\it Y. Yu, L. Wen} and {\it S. Li} [Appl. Math. Comput. 191, No. 2, 543--549 (2007)]. In the end of the article, a numerical example is considered.

65R20Integral equations (numerical methods)
34K40Neutral functional-differential equations
Full Text: DOI
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