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

Gui-Lai Zhang (), Zhi-Yong Zhu, Yu-Chen Wang and Chao Liu
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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

Mathematics, 2024, vol. 12, issue 19, 1-30

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.

Keywords: impulsive discrete Runge–Kutta method; impulsive continuous Runge–Kutta method; Lipschitz condition; convergence; asymptotical stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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