Asymptotical stability of Runge–Kutta methods for nonlinear impulsive differential equations
In this paper, asymptotical stability of the exact solutions of nonlinear impulsive ordinary differential equations is studied under Lipschitz conditions. Under these conditions, asymptotical stability of Runge–Kutta methods is studied by the theory of Padé approximation. And two simple examples are...
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| Published in: | Advances in difference equations Vol. 2020; no. 1; pp. 1 - 12 |
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| Abstract | In this paper, asymptotical stability of the exact solutions of nonlinear impulsive ordinary differential equations is studied under Lipschitz conditions. Under these conditions, asymptotical stability of Runge–Kutta methods is studied by the theory of Padé approximation. And two simple examples are given to illustrate the conclusions. |
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| AbstractList | In this paper, asymptotical stability of the exact solutions of nonlinear impulsive ordinary differential equations is studied under Lipschitz conditions. Under these conditions, asymptotical stability of Runge–Kutta methods is studied by the theory of Padé approximation. And two simple examples are given to illustrate the conclusions. Abstract In this paper, asymptotical stability of the exact solutions of nonlinear impulsive ordinary differential equations is studied under Lipschitz conditions. Under these conditions, asymptotical stability of Runge–Kutta methods is studied by the theory of Padé approximation. And two simple examples are given to illustrate the conclusions. |
| ArticleNumber | 42 |
| Author | Zhang, Gui-Lai |
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Math.19781847548952075610.1007/BF01932026 – reference: RanX.J.LiuM.Z.ZhuQ.Y.Numerical methods for impulsive differential equationMath. Comput. Model.2008484655243132210.1016/j.mcm.2007.09.010 – reference: LiuX.ZhangG.L.LiuM.Z.Analytic and numerical exponential asymptotic stability of nonlinear impulsive differential equationsAppl. Numer. Math.2014814049321217410.1016/j.apnum.2013.12.009 – reference: SamoilenkoA.M.PerestyukN.A.ChapovskyY.Impulsive Differential Equations1995SingaporeWorld Scientific10.1142/2892 – reference: WuK.N.NaM.Y.WangL.M.DingX.H.WuB.Y.Finite-time stability of impulsive reaction–diffusion systems with and without time delayAppl. Math. Comput.2019363398202007149669 – reference: WuK.Z.DingX.H.WangL.M.Stability and stabilization of impulsive stochastic delay difference equationsDiscrete Dyn. Nat. Soc.20102010260032810.1155/2010/5920361189.93140 – reference: LiangH.LiuM.Z.Extinction and permanence of the numerical solution of a two-prey one-predator system with impulsive effectInt. J. Comput. Math.20148813051325278356610.1080/00207160.2010.504829 – reference: LiuM.Z.LiangH.YangZ.W.Stability of Runge–Kutta methods in the numerical solution of linear impulsive differential equationsAppl. Math. Comput.200719234635723856001193.65121 – reference: WuK.Z.DingX.H.Impulsive stabilization of delay difference equations and its application in Nicholson’s blowflies modelsAdv. Differ. Equ.20122012294873210.1186/1687-1847-2012-881302.39028 – reference: WuK.Z.DingX.H.Stability and stabilization of impulsive stochastic delay differential equationsMathematical Problems in Engineering 2012201210.1155/2012/176375 – reference: DekkerK.VerwerJ.G.Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations1984AmsterdamNorth-Holland0571.65057 – reference: LiangH.SongM.H.LiuM.Z.Stability of the analytic and numerical solutions for impulsive differential equationsAppl. Numer. Math.20116111031113284213310.1016/j.apnum.2010.12.005 – reference: LakshmikanthamV.BainovD.D.SimeonovP.S.Theory of Impulsive Differential Equations1989SingaporeWorld Scientific10.1142/0906 – reference: Mil’manV.D.MyshkisA.D.On the stability of motion in the presence of impulsesSib. Math. J.196012332371260281358.34022 – reference: LiuX.ZengY.M.Linear multistep methods for impulsive delay differential equationsAppl. Numer. Math.20183215535633732397 – reference: WenL.P.YuY.S.The analytic and numerical stability of stiff impulsive differential equations in Banach spaceAppl. Math. Lett.20112417511757280301910.1016/j.aml.2011.04.042 – reference: BainovD.D.SimeonovP.S.Systems with Impulsive Effect: Stability, Theory and Applications1989ChichesterEllis Horwood – reference: SongM.H.YangZ.W.LiuM.Z.Stability of θ-methods for advanced differential equations with piecewise continuous argumentsComput. Math. Appl.20054912951301214948110.1016/j.camwa.2005.02.002 – reference: ButcherJ.C.The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods1987New YorkWiley0616.65072 – reference: BainovD.D.SimeonovP.S.Impulsive Differential Equations: Asymptotic Properties of the Solutions1995SingaporeWorld Scientific10.1142/2413 – reference: WangQ.QiuS.Oscillation of numerical solution in the Runge–Kutta methods for equation x′(t)=ax(t)+a0x([t])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x'(t) = ax(t) + a_{0} x([t])$\end{document}Acta Math. Appl. Sin. Engl. 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| SubjectTerms | Analysis Asymptotic methods Asymptotic properties Asymptotical stability Difference and Functional Equations Differential equations Exact solutions Functional Analysis Impulsive differential equation Lipschitz condition Mathematical analysis Mathematics Mathematics and Statistics Nonlinear equations Ordinary Differential Equations Pade approximation Partial Differential Equations Runge-Kutta method Stability |
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| Title | Asymptotical stability of Runge–Kutta methods for nonlinear impulsive differential equations |
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