Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients

Year 2022, Volume: 5 Issue: 3, 107 - 116, 30.09.2022

Hefan Yin , Qi Wang

https://doi.org/10.32323/ujma.1105072

Abstract

The paper discusses the analytical stability and numerical stability of differential equations with piecewise constant arguments with matrix coefficients. Firstly, the Runge-Kutta method is applied to the equation and the recurrence relationship of the numerical solution of the equation is obtained. Secondly, it is proved that the Runge-Kutta method can preserve the convergence order. Thirdly, the stability conditions of the numerical solution under different matrix coefficients are given by Pad$\acute{e}$ approximation and order star theory. Finally, the conclusions are verified by several numerical experiments.

Keywords

Runge-Kutta methods, analytical stability, numerical stability

Supporting Institution

Natural Science Foundation of Guangdong Province

Project Number

2017A030313031

Thanks

This study is supported by the Natural Science Foundation of Guangdong Province with the project number 2017A030313031.

References

• [1] L. E. Shaikhet, Behavior of solution of stochastic delay differential equation with additive fading perturbations, Appl. Math. Lett., (2021), Article ID: 106640, 9 pages.
• [2] N. Senu, K. C. Lee, A. Ahmadian, S. N. I. Ibrahim, Numerical solution of delay differential equation using two-derivative Runge-Kutta type method with Newton interpolation, Alex. Eng. J., 61 (2022), 5819-5835.
• [3] K. E. M. Church, G. W. Duchesne, Rigorous continuation of periodic solutions for impulsive delay differential equations, Appl. Math. Comput., 415 (2022), Article ID: 126733, 27 pages.
• [4] L. Berezansky, E. Braverman, Solution estimates for linear differential equations with delay, Appl. Math. Comput., 372 (2020), Article ID: 124962, 15 pages.
• [5] C. Jamilla, R. Mendoza, I. Mez, Solutions of neutral delay differential equations using a generalized Lambert W function, Appl. Math. Comput., 382 (2020), Article ID: 125334, 17 pages.
• [6] G. Philipp, W. Marcus, A differential equation with state-dependent delay from cell population biology, J. Differ. Equ., 260(7) (2016), 6176-6200.
• [7] F. Karakoc, Asymptotic behavior of a population model with piecewise constant argument, Appl. Math. Lett., 70 (2017), 7-13.
• [8] K. F. Owusu, E. F. D. Goufo, S. Mugisha, Modelling intracellular delay and therapy interruptions within Ghanaian HIV population, Adv. Differ. Equ., 2020 (2020), 19 pages.
• [9] C. Monica, M. Pitchaimani, Geometric stability switch criteria in HIV-1 infection delay model, J. Nonlinear Sci., 29 (2019), 163-181.
• [10] O. Matte, Continuity properties of the semi-group and its integral kernel in non-relativistic QED, Rev. Math. Phys., 28(5) (2016), 1-76.
• [11] M. Dehghan, F. Shakeri, The use of the decomposition procedure of Adomain for solving a delay differential equation arising in electrodynamics, Phys. Scr., 78(6) (2008), Article ID: 065004, 11 pages.
• [12] Y. H. Chen, H. Y. Yu, X. Y. Meng, X. L. Xie, M. Z. Hou, J. Chevallier, Numerical solving of the generalized Black-Scholes differential equation using Laguerre neural network, Digit. Signal Process., 112(5) (2021), Article ID: 103003, 11 pages.
• [13] M. Akhmet, D. A. Cincin, M. Tleubergenova, Z. Nugayeva, Unpredictable oscillations for Hopfield-type neutral networks with delayed and advanced arguments, Mathematics, 9(5) (2021), 19 pages.
• [14] L. Zhang, M. H. Song, Convergence of the Euler method of stochastic differential equations with piecewise continuous arguments, Abstr. Appl. Anal., 2012 (2012), Article ID: 643783, 16 pages.
• [15] S. R. Raj, M. Saradha, Solving hybrid fuzzy fractional differential equations by improved Euler method, Math. Theory Model., 5(5) (2015), 106-117.
• [16] Q. Wang, The numerical asymptotically stability of a linear differential equation with piecewise constant arguments of mixed type, Acta Appl. Math., 146 (2016), 145-161.
• [17] J. F. Gao, Numerical oscillation and non-oscillation for differential equation with piecewise continuous arguments of mixed type, Appl. Math. Comput., 299 (2017), 16-27.
• [18] W. S. Wang, Stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations, Appl. Math. Comput., 219 (2013), 4590-4600.
• [19] G. L. Zhang, Stability of Runge-Kutta methods for linear impulsive delay differential equations with piecewise constant arguments, J. Comput. Appl. Math., 297 (2016), 41-50.
• [20] H. Liang, M. Z. Liu, Z. W. Yang, Stability analysis of Runge-Kutta methods for systems u0(t) = Lu(t)+Mu([t]), Appl. Math. Comput., 228 (2014), 463-476.
• [21] J. Wiener, General Solutions of Differential Equations, World Scientific, Singapore, 1993.
• [22] K. Dekker, J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, 1984.
• [23] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, Wiley, New York, 1987.
• [24] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential Algebraic Problems, Springer, New York, 1996.