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Convergence, consistency and zero stability of impulsive one-step numerical methods

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  • Zhang, Gui-Lai

Abstract

Impulsive one-step numerical methods are defined in the present paper, especially, a common and widely used numerical form generalised from Runge-Kutta methods defined as impulsive Runge-Kutta methods. And it is proved that a consistent and zero-stable method thus convergent. Moreover, it is also proved that an impulsive one-step numerical method is convergent of order p if the corresponding method is pth order. Another equivalent form of impulsive one-step numerical methods are also introduced. In addition, numerical experiments are provided to illustrate the advantage of impulsive Runge-Kutta methods.

Suggested Citation

  • Zhang, Gui-Lai, 2022. "Convergence, consistency and zero stability of impulsive one-step numerical methods," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    • Handle: RePEc:eee:apmaco:v:423:y:2022:i:c:s0096300322001035
    DOI: 10.1016/j.amc.2022.127017
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    References listed on IDEAS

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    1. Zhang, Gui-Lai & Song, Ming-Hui, 2019. "Impulsive continuous Runge–Kutta methods for impulsive delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 160-173.
    2. Liu, X. & Zeng, Y.M., 2018. "Linear multistep methods for impulsive delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 555-563.
    3. Liming Wang & Baoqing Yang & Xiaohua Ding & Kai-Ning Wu, 2015. "Boundedness of Stochastic Delay Differential Systems with Impulsive Control and Impulsive Disturbance," Mathematical Problems in Engineering, Hindawi, vol. 2015, pages 1-8, March.
    4. Xiaotai Wu & Litan Yan & Wenbing Zhang & Liang Chen, 2012. "Exponential Stability of Impulsive Stochastic Delay Differential Systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 2012, pages 1-15, January.
    5. X. Liu & M. H. Song & M. Z. Liu, 2012. "Linear Multistep Methods for Impulsive Differential Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2012, pages 1-14, May.
    6. Kaining Wu & Xiaohua Ding, 2012. "Stability and Stabilization of Impulsive Stochastic Delay Differential Equations," Mathematical Problems in Engineering, Hindawi, vol. 2012, pages 1-16, May.
    7. Kaining Wu & Xiaohua Ding & Liming Wang, 2010. "Stability and Stabilization of Impulsive Stochastic Delay Difference Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2010, pages 1-15, March.
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    Cited by:

    1. Gui-Lai Zhang & Zhi-Yong Zhu & Yu-Chen Wang & Chao Liu, 2024. "Impulsive Discrete Runge–Kutta Methods and Impulsive Continuous Runge–Kutta Methods for Nonlinear Differential Equations with Delayed Impulses," Mathematics, MDPI, vol. 12(19), pages 1-30, September.
    2. Gui-Lai Zhang & Chao Liu, 2024. "Two Schemes of Impulsive Runge–Kutta Methods for Linear Differential Equations with Delayed Impulses," Mathematics, MDPI, vol. 12(13), pages 1-17, July.

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