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Numerical methods for impulsive differential equation. (English) Zbl 1145.65317

Summary: In this paper, the asymptotical stability of the numerical methods with the constant stepsize for impulsive differential equation

x ˙(t)=αx,tk,t>0Δx=σx,t=kx(0+0)=x 0 ,

where a0,β,x 0 ,1+β0,k, are investigated. The asymptotical stability conditions of the analytic solution of this equation and the numerical solutions are obtained. Finally, some experiments are given.

65L05Initial value problems for ODE (numerical methods)
[1]Bainov, D. D.; Simenov, P. S.: Systems with impulse effect stability theory and applications, (1989) · Zbl 0683.34032
[2]Bainov, D. D.; Kostadinov, S. I.; Van Minh, N.; Zabreiko, P. P.: A topological classification of differential equations with impulse effect, Tamkang J. Math. 25, 15-27 (1994) · Zbl 0805.34041
[3]Bainov, D. D.; Simenov, P. S.: Systems with impulse effect, stability theory and applications, Systems with impulse effect, stability theory and applications 40 (1989)
[4]Borisenko, S. D.; Kosolapov, V. I.; Obolenskii, Yu.A.: Stability of processes under continuations and discrete disturbances, (1988)
[5]Kulev, G.; Bainov, D. D.: On the stability of systems with impulsive by sirect method of Lyapunov, J. math. Anal. appl. 140, 324-340 (1989) · Zbl 0681.34042 · doi:10.1016/0022-247X(89)90067-X
[6]Bainov, D. D.; Kulev, G.: Application of Lyapunov’s functions to the investigation of global stability of solutions of system with impulses, Appl. anal. 26, No. 1, 255-270 (1988)
[7]Randelovic, B. M.; Stefanovic, L. V.; Dankovic, B. M.: Numerical solution of impulsive differential equations, Facta univ. Ser. math. Inform. 15, 101-111 (2000) · Zbl 1052.65063
[8]Butcher, J. C.: The numerical analysis of ordinary differential equations: Runge–Kutta methods, (1987) · Zbl 0616.65072
[9]Dekker, K.; Verwer, J. G.: Stability of Runge–Kutta methods for stiff nonlinear differential equations, (1984)
[10]Jianhua, Shen: New maximum principles for first-order impulsive boundary value problems, Appl. math. Lett. 16, 105-112 (2003) · Zbl 1027.34033 · doi:10.1016/S0893-9659(02)00151-9
[11]Hairer, E.; Nøsett, S. P.; Wanner, G.: Solving ordinary differential equations II, stiff and differential algebraic problems, (1993)
[12]Akhmetov, M. U.; Zafer, A.: Successive approximation method for quasilinear impulsive differential equations with control, Appl. math. Lett. 13, 99-105 (2000) · Zbl 1125.93349 · doi:10.1016/S0893-9659(00)00040-9
[13]Lambert, J. D.: Computational methods in ordinary differential equations, (1972)
[14]Lakshmikantham, V.; Bainov, D. D.; Simenov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[15]Dâââonofrio, Alberto: On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. math. Lett. 18, 729-732 (2005) · Zbl 1064.92041 · doi:10.1016/j.aml.2004.05.012
[16]Nieto, J. J.: Impulsive resonance periodic problems of first order, Appl. math. Lett. 15, 489-493 (2002) · Zbl 1022.34025 · doi:10.1016/S0893-9659(01)00163-X
[17]Samoilenko, A. M.; Perestyuk, N. A.: Differential equations with impulse effect, (1987)
[18]Song, M. H.; Yang, Z. W.; Liu, M. Z.: Stability of θ-methods for advanced differential equations with piecewise continuous arguments, J. comput. Math. appl. 49, 1295-1301 (2005) · Zbl 1082.65078 · doi:10.1016/j.camwa.2005.02.002
[19]Xu, X. Y.; Li, J. K.; Xu, G. L.: Introduction to Padé approximation, (1990)
[20]Fu, X. L.; Yan, B. Q.: Introduction to the impulsive differential system, (2005)