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.


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
Full Text: DOI


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