## Runge-Kutta methods for the multi-pantograph delay equation.(English)Zbl 1070.65060

The multi-pantograph equation is a delay differential equation of the form $$u'(t) = \lambda u(t) + \mu_1 u(q_1 t) + \mu_2 u(q_2 t) + \cdots + \mu_l u(q_l t)$$, with given initial value $$u(0)$$. It is assumed that $$0<q_l < \cdots < q_2 < q_1 <1$$ and that $$\lambda$$ and $$\mu_k (k=1,2,\dots,l$$) are complex numbers. If $$\text{Re} \lambda <0$$ and $$\sum_{k=1}^l | \mu_k| < \lambda$$, then it is known [cf. A. Iserles and J. Terjéki, J. Lond. Math. Soc., II. Ser. 51, No. 3, 559–572 (1995; Zbl 0832.34080)] that the exact solution tends to zero as $$t\to \infty$$.
It is interesting to consider when a numerical approximation scheme satisfies a corresponding asymptotic stability property. This paper is concerned with formulating Runge-Kutta methods to solve the multi-pantograph equation and to establish sufficient conditions for asymptotic stability.

### MSC:

 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34K20 Stability theory of functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)

Zbl 0832.34080
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### References:

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