Abstract
Numerical oscillation of Runge-Kutta methods for differential equations with piecewise constant arguments is considered in this paper. The conditions of oscillation for the Runge-Kutta methods are obtained. It is proven that the numerical oscillation on the integer nodes are equivalent to the numerical oscillation on the any nodes and oscillation of the analytic solution is preserved by the Runge-Kutta methods. Moreover, the relationship between stability and oscillation is discussed for analytic solution and numerical solution, respectively. At last, several numerical simulations are carried out to support the theoretical analysis of the research.
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References
Akhmet, M.U., Arugaslan, D., Yllmaz, E.: Stability in Cellular Neural Networks with a Piecewise Constant Argument. J. Comput. Appl. Math. 233, 2365–2373 (2010)
Akhmet, M.U., Arugaslan, D., Yllmaz, E.: Method of Lyapunov Functions for Differential Equations with Piecewise Constant Delay. J. Comput. Appl. Math. 235, 4554–4560 (2011)
Dimbour, W.: Almost Automorphic Solutions for Differential Equations with Piecewise Constant Argument in a Banach Space. Nonlinear Anal. 74, 2351–2357 (2011)
Akhmet, M.U., Arugaslan, D., Yllmaz, E.: Stability Analysis of Recurrent Neural Networks with Piecewise Constant Argument of Generalized Type. Neural Networks 23, 805–811 (2010)
Fu, X.L., Li, X.D.: Oscillation of Higher Order Impulsive Differential Equations of Mixed Type with Constant Argument at Fixed Time. Math. Comput. Model. 48, 776–786 (2008)
Luo, Z.G., Shen, J.H.: New Results on Oscillation for Delay Differential Equations with Piecewise Constant Argument. Comput. Math. Appl. 45, 1841–1848 (2003)
Shen, J.H., Stavroulakis, I.P.: Oscillatory and Nonoscillatory Delay Equations with Piecewise Constant Argument. J. Math. Anal. Appl. 248, 385–401 (2000)
Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)
Song, M.H., Liu, X.: The Improved Linear Multistep Methods for Differential Equations with Piecewise Continuous Arguments. Appl. Math. Comput. 217, 4002–4009 (2010)
Liu, M.Z., Ma, S.F., Yang, Z.W.: Stability Analysis of Runge-Kutta Methods for Unbounded Retarded Differential Equations with Piecewise Continuous Arguments. Appl. Math. Comput. 191, 57–66 (2007)
Song, M.H., Yang, Z.W., Liu, M.Z.: Stability of θ-Methods for Advanced Differential Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 49, 1295–1301 (2005)
Dai, H.Y., Liu, M.Z.: Mean-square Stability of Stochastic Differential Equations with Piecewise Continuous Arguments. Heilongjiang Univ. J. Nat. Sci. 25, 625–629 (2008)
Liu, M.Z., Gao, J.F., Yang, Z.W.: Oscillation Analysis of Numerical Solution in the θ-Methods for Equation x’(t)+ax(t)+a 1 x([t-1])=0. Appl. Math. Comput. 186, 566–578 (2007)
Liu, M.Z., Gao, J.F., Yang, Z.W.: Preservation of Oscillations of the Runge-Kutta Method for Equation x’(t)+ax(t)+a 1 x([t-1])=0. Comput. Math. Appl. 58, 1113–1125 (2009)
Gao, J.F., Liu, M.Z.: Numerical Oscillations of the θ-Method for Advanced Delay Differential Equations with Piecewise Continuous Arguments. Heilongjiang Univ. J. Nat. Sci. 25, 196–203 (2008)
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Wang, Q., Wen, J., Fu, F. (2012). Oscillation of Runge-Kutta Methods for a Scalar Differential Equation with One Delay. In: Liu, C., Wang, L., Yang, A. (eds) Information Computing and Applications. ICICA 2012. Communications in Computer and Information Science, vol 307. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34038-3_46
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DOI: https://doi.org/10.1007/978-3-642-34038-3_46
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