Asymptotical stability of Runge–Kutta methods for nonlinear impulsive differential equations

Zhang, Gui-Lai

doi:10.1186/s13662-019-2473-x

Cited by 2 publications

(2 citation statements)

References 25 publications

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“…Remark 3. Most of the literature on the stability analysis of the numerical method for impulsive differential equations is based on the classic Lipschitz or one-sided Lipschitz conditions in the sense of the standard inner product norm (see [9,13,20]). If the value of the one-sided Lipschitz constant of the problem is very large (see problem (42)), these classic stability theories will fail.…”

Section: Stability and Asymptotic Stability Of Multistage One-stepmentioning

confidence: 99%

“…In [12], Mei proposed a new algorithm based on the reproducing kernel method and least square method for solving nonlinear IDEs. Zhang gave the asymptotic stability of the analytic solution of nonlinear impulsive differential equations under Lipschitz condition and studied the asymptotic stability of Runge-Kutta methods by using Pade approximation theory in [13].…”

Section: Introductionmentioning

confidence: 99%

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Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

Long

2021

Mathematical Problems in Engineering

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In this paper, we first introduce the problem class K p μ , λ , ζ with respect to the initial value problems of nonlinear impulsive differential equations in Banach spaces. The stability and asymptotic stability results of the analytic solution of the problem class K p μ , λ , ζ are obtained. Then, the numerical stability and asymptotic stability conditions of multistage one-step multiderivative methods are also given. Two numerical experiments are given to confirm the theoretical results in the end.

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Section: Stability and Asymptotic Stability Of Multistage One-stepmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

Long

2021

Mathematical Problems in Engineering

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Impulsive Discrete Runge–Kutta Methods and Impulsive Continuous Runge–Kutta Methods for Nonlinear Differential Equations with Delayed Impulses

Zhang,

Zhu,

Wang

et al. 2024

Mathematics

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In this paper, we study the asymptotical stability of the exact solutions of nonlinear impulsive differential equations with the Lipschitz continuous function f(t,x) for the dynamic system and for the impulsive term Lipschitz continuous delayed functions Ik. In order to obtain numerical methods with a high order of convergence and that are capable of preserving the asymptotical stability of the exact solutions of these equations, impulsive discrete Runge–Kutta methods and impulsive continuous Runge–Kutta methods are constructed, respectively. For these different types of numerical methods, different convergence results are obtained and the sufficient conditions for asymptotical stability of these numerical methods are also obtained, respectively. Finally, some numerical examples are provided to confirm the theoretical results.

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Asymptotical stability of Runge–Kutta methods for nonlinear impulsive differential equations

Cited by 2 publications

References 25 publications

Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

Impulsive Discrete Runge–Kutta Methods and Impulsive Continuous Runge–Kutta Methods for Nonlinear Differential Equations with Delayed Impulses

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