Numerical stability of coupled differential equation with piecewise constant arguments

Qi Wang

Qi Wang School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China

Keywords: Coupled differential equation; Piecewise constant arguments; Linear method; Stability.

Abstract

This paper deals with the stability of numerical solutions for a coupled differential equation with piecewise constant arguments. A sufficient condition such that the system is asymptotically stable is derived. Furthermore, when the linear θ-method is applied to this system, it is shown that the linear θ-method is asymptotically stable if and only if 1/2<θ≤1. Finally, some numerical experiments are given.

Downloads

Download data is not yet available.

References

[1] I. Ozturk, F. Bozkurt, “Stability analysis of a population model with piecewise constant arguments,” Nonlinear Analysis: R.W.A. 12 (2011) 1532-1545.
[2] L.F. Wang, H.Q. Wu, D.Y. Liu et al., “Lur'e Postnikov Lyapunov functional technique to global Mittag-Leffler stability of fractional-order neural networks with piecewise constant argument,” Neurocomputing 302 (2018) 23-32.
[3] Y. Muroya, “New contractivity condition in a population model with piecewise constant arguments,” J. Math. Anal. Appl. 346 (2008) 65-81.
[4] X.L. Fu, X.D. Li, “Oscillation of higher order impulsive differential equations of mixed type with constant argument at fixed time,” Math. Comput. Model. 48 (2008) 776-786.
[5] L.L. Zhang, H.X. Li, “Weighted pseudo almost periodic solutions of second order neutral delay differential equations with piecewise constant argument,” Comput. Math. Appl. 62 (2011) 4362-4376.
[6] K.S. Chiu, “Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument,” Acta Appl. Math. 151 (2017) 199-226.
[7] S. Kartal, “Flip and Neimark-Sacker bifurcation in a differential equation with piecewise constant arguments model,” J. Differ. Equ. Appl. 23 (2017) 763-778.
[8] S. Kartal, “Multiple bifurcations in an early brain tumor model with piecewise constant arguments,” Int. J. Biomath. 11 (2018) 1-19.
[9] M. Pinto, “Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments,” Math. Comput. Model. 49 (2009) 1750-1758.
[10] F. Karakoc, “Asymptotic behaviour of a population model with piecewise constant argument,” Appl. Math. Lett. 70 (2017) 7-13.
[11] J. Wiener, “Generalized Solutions of Functional Differential Equations,” World Scientific, Singapore, 1993.
[12] M.Z. Liu, M.H. Song, Z.W. Yang, “Stability of Runge–Kutta methods in the numerical solution of equation ,”. J. Comput. Appl. Math. 166 (2004) 361-370.
[13] M.Z. Liu, J.F. Gao, Z.W. Yang, “Oscillation analysis of numerical solution in the -methods for equation ,” Appl. Math. Comput. 186 (2007) 566-578.
[14] M.Z. Liu, J.F. Gao, Z.W. Yang, “Preservation of oscillations of the Runge-Kutta method for equation ,” Comput. Math. Appl. 58 (2009) 1113-1125.
[15] L.P. Wen, Y.X. Yu, S.F. Li, “Dissipativity of linear multistep methods for nonlinear differential equations with piecewise delays,” Math. Numer. Sinica 28 (2006) 67-74.
[16] C. Li, C.J. Zhang, “Block boundary value methods applied to functional differential equations with piecewise continuous arguments,” Appl. Numer. Math. 115 (2017) 214-224.
[17] Y.L. Lu, M.H. Song, M.Z. Liu, “Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments,” J. Comput. Appl. Math. 317 (2017) 55-71.
[18] J.F. Gao, “Numerical oscillation and non-oscillation for differential equation with piecewise continuous arguments of mixed type,” Appl. Math. Comput. 299 (2017) 16-27.
[19] X.Y. Li, H.X. Li, B.Y. Wu, “Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments,” Appl. Math. Comput. 349 (2019) 304-313.
[20] H. Liang, M.Z. Liu, Z.W. Yang, “Stability analysis of Runge-Kutta methods for systems ,” Appl. Math. Comput. 228 (2014) 463-476.
[21] A.R. Aftabizadeh, J. Wiener, “Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument,” Appl. Anal., 26 (1988) 327-333.
[22] M.H. Song, Z.W. Yang, M.Z. Liu, “Stability of -methods for advanced differential equations with piecewise continuous arguments,” Comput. Math. Appl. 49 (2005) 1295-1301.
[23] R.A. Horn, C.R. Johnson, “Matrix Analysis,” Cambridge University Press, Cambridge, 1985.
[24] A. Bellen, N. Guglielmi, L. Torelli, “Asymptotic stability properties of -methods for the pantograph equation,” Appl. Numer. Math. 24 (1997) 279-293.