×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ajello, W.G.; Freedman, H.I.; Wu, J., A model of stage structured population growth with density dependent time delay, SIAM J. appl. math., 52, 855-869, (1992) · Zbl 0760.92018
[2] Buhmann, M.D.; Iserles, A., Stability of the discretized pantograph differential equation, Math. comp., 60, 575-589, (1993) · Zbl 0774.34057
[3] Bellen, A.; Guglielmi, N.; Torelli, L., Asymptotic stability properties of θ-methods for pantograph equation, Appl. numer. math., 24, 279-293, (1997) · Zbl 0878.65064
[4] Butcher, J.C., The numerical analysis of ordinary differential equation, (1987), John Wiley New York · Zbl 0616.65072
[5] Diekmman, O., Delay equation: function, complex and nonlinear analysis, (1995), Springer-Verlag New York
[6] Ding, X.; Liu, M., Convergence aspects of step-parallel iteration of runge – kutta methods for delay differential equation, Bit, 42, 3, 508-518, (2002) · Zbl 1020.65041
[7] Iserles, A.; Terijeki, J., Stability and asymptotic stability of functional-differential equation, J. London math soc., 2, 559-572, (1995) · Zbl 0832.34080
[8] Iserles, A., Numerical analysis of delay differential equation with variable delay, Ann. numer. math., 1, 133-152, (1994) · Zbl 0828.65083
[9] Fox, L.; Mayers, D.F.; Ockendon, J.A.; Tayler, A.B., On a functional differential equation, J. inst. math. appl., 8, 271-307, (1971) · Zbl 0251.34045
[10] Guglielmi, N., Geometric proofs numerical stability for delay equations, IMA J. numer. anal., 21, 439-450, (2001) · Zbl 0976.65077
[11] In’t Hout, K.J., On the stability of adaptations of runge – kutta methods to systems of delay differential equations, Appl. numer. math., 22, 237-250, (1996) · Zbl 0867.65045
[12] Liu, M.Z.; Spijker, M., The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. numer. anal., 10, 31-48, (1990) · Zbl 0693.65056
[13] Liu, Y., Numerical investigation of the pantograph equation, Appl. numer. math., 24, 309-317, (1997) · Zbl 0878.65065
[14] Ockendon, J.R.; Tayler, A.B., The dynamics of a current collection system for an electric locomotive, Proc. roy. London ser. A, 322, 447-468, (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.