Properties of analytic solution and numerical solution of multi-pantograph equation. (English) Zbl 1059.65060

This paper contains the properties of analytical and numerical solutions of the multi-pantograph equation \(u'(t)=\lambda u(t)+\sum_{i=1}^l u_i u(q_i(t))\). The sufficient condition of the asymptotic stability, the existence and the uniqueness of the analytical solution of the above equation are obtained. Numerical examples are provided to show that the properties of the \(\theta\)-methods are asymptotically stable if \(\frac12<\theta\leq 1\).


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
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[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. Comput., 60, 575-589 (1993) · Zbl 0774.34057
[3] Bellen, A.; Guglielmi, N.; Torelli, L., Asymptotic stability properties of theta-methods for pantograph equation, Appl. Numer. Math., 24, 279-293 (1997) · Zbl 0878.65064
[4] Diekmman, O., Delay Equation: Function-, Complex and Nonlinear Analysis (1995), Springer-Verlag: Springer-Verlag New York
[5] 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
[6] 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
[7] Guglielmi, N., Geometric proofs numerical stability for delay equations, IMA J. Anal., 21, 439-450 (2001) · Zbl 0976.65077
[8] Hout, I., 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
[9] Iserles, A., Numerical analysis of delay differential equation with variable delay, Ann. Numer. Math., 1, 133-152 (1994) · Zbl 0828.65083
[10] Li, D.; Liu, M., The properties of exact solution of multi-pantograph delay differential equation, J. Harbin Inst. Technol., 3, 1-3 (2000), (Chinese) · Zbl 1087.34536
[11] 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
[12] Ockendon, J. R.; Tayler, A. B., The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. London Ser. A, 322, 447-468 (1971)
[13] Palka, B. P., An Introduction to Complex Function Theory (1990), Springer-Verlag: Springer-Verlag New York
[14] Qiu, L.; Mitsui, T.; Kuang, J.-x, The numerical stability of the \(θ\)-methods for delay differential equations with many variable delay, JCM, 17, 5, 523-532 (1999) · Zbl 0942.65087
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