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

Author

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  • Gui-Lai Zhang

    (College of Sciences, Northeastern University, Shenyang 110819, China)

  • Zhi-Yong Zhu

    (College of Sciences, Northeastern University, Shenyang 110819, China)

  • Yu-Chen Wang

    (College of Sciences, Northeastern University, Shenyang 110819, China)

  • Chao Liu

    (College of Sciences, Northeastern University, Shenyang 110819, China)

Abstract

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 I k . 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.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:3002-:d:1486602
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    References listed on IDEAS

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    1. Zhang, G.L. & Song, M.H., 2015. "Asymptotical stability of Runge–Kutta methods for advanced linear impulsive differential equations with piecewise constant arguments," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 831-837.
    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.
    3. Bellen, Alfredo & Zennaro, Marino, 2003. "Numerical Methods for Delay Differential Equations," OUP Catalogue, Oxford University Press, number 9780198506546.
    4. Zhang, G.L. & Song, Minghui & Liu, M.Z., 2015. "Asymptotical stability of the exact solutions and the numerical solutions for a class of impulsive differential equations," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 12-21.
    5. Zhang, Gui-Lai, 2022. "Convergence, consistency and zero stability of impulsive one-step numerical methods," Applied Mathematics and Computation, Elsevier, vol. 423(C).
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