Stability analysis of Runge-Kutta methods for unbounded retarded differential equations with piecewise continuous arguments. (English) Zbl 1193.65122

Liu, M. Z.; Ma, S. F.; Yang, Z. W.

Appl. Math. Comput. 191, No. 1, 57-66 (2007).

Summary: This paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments (EPCA) \(u'(t) =f(t,u(t)\), \(u([t/N]))\), \(t > 0\), \(u(0) = u_0\), where \(N\in\mathbb{Z}_+\). A sufficient condition that the differential equations are asymptotically stable is derived. This paper is also concerned with the stability analysis of the Runge-Kutta methods for equation \(u'(t) = au(t) + bu([t/N])\), \(t\geq 0\), \(u(0) = u_0\), where \(N\in\mathbb{Z}_+\). The conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given.

MSC:

65L06

Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

the unbounded retarded differential equation; piecewise continuous arguments; asymptotic stability

Software:

RODAS

References:

[1]	Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987), Wiley: Wiley New York · Zbl 0616.65072
[2]	Cooke, K. L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl., 99, 265-297 (1984) · Zbl 0557.34059
[3]	Dekker, K.; Verwer, J. G., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (1984), North-Holland: North-Holland Amsterdam · Zbl 0571.65057
[4]	Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations II, Stiff and Differential Algebraic Problems (1993), Springer-Verlag: Springer-Verlag New York
[5]	Iserles, A.; Nørsett, S. P., Order stars and rational approximations to exp \((z)\), Appl. Numer. Math., 5, 63-70 (1989) · Zbl 0674.65043
[6]	Liu, M. Z.; Song, M. H.; Yang, Z. W., Stability of Runge-Kutta methods in the numerical solution of equation \(u^\prime(t) = au(t) + a_0 u([t])\), J. Comput. Appl. Math., 166, 361-370 (2004) · Zbl 1052.65070
[8]	Shah, S. M.; Wiener, J., Advanced differential equations with piecewise constant argument deviations, Int. J. Math. Sci., 6, 671-703 (1983) · Zbl 0534.34067
[9]	Song, M. H.; Yang, Z. W.; Liu, M. Z., Stability of \(θ\)-methods advanced differential equations with piecewise continuous argument, Comput. Math. Appl., 49, 1205-1301 (2005) · Zbl 1082.65078
[10]	Wanner, G.; Hairer, E.; Nørsett, S. P., Order stars and stability theorems, BIT, 18 (1978) · Zbl 0444.65039
[11]	Wiener, J., Differential equations with piecewise constant delays, (Lakshmikantham, V., Trends in the Theory and Practice of Nonlinear Differential Equations (1983), Marcel Dekker: Marcel Dekker New York), 547-552
[12]	Wiener, J., Generalised Solutions of Differential Equations (1993), World Scientific: World Scientific Singapore
[13]	Yang, Z. W.; Liu, M. Z.; Song, M. H., Stability of Runge-Kutta methods in the numerical solution of equation \(u^\prime(t) = au(t) + a_0 u([t]) + a_1 u([t - 1])\), Appl. Math. Comput., 162, 37-50 (2005) · Zbl 1063.65070