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

Hefan Yin; Qi Wang

doi:10.32323/ujma.1105072

EN

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

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.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Hefan Yin ^*
0000-0001-7182-3272
China

Qi Wang
0000-0003-3578-2551
China

Publication Date

September 30, 2022

Submission Date

April 18, 2022

Acceptance Date

September 15, 2022

Published in Issue

Year 2022 Volume: 5 Number: 3

DOI

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

IZ

https://izlik.org/JA62ZA85JW

APA

Yin, H., & Wang, Q. (2022). Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients. Universal Journal of Mathematics and Applications, 5(3), 107-116. https://doi.org/10.32323/ujma.1105072

AMA

1.Yin H, Wang Q. Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients. Univ. J. Math. Appl. 2022;5(3):107-116. doi:10.32323/ujma.1105072

Chicago

Yin, Hefan, and Qi Wang. 2022. “Numerical Stability of Runge-Kutta Methods for Differential Equations With Piecewise Constant Arguments With Matrix Coefficients”. Universal Journal of Mathematics and Applications 5 (3): 107-16. https://doi.org/10.32323/ujma.1105072.

EndNote

Yin H, Wang Q (September 1, 2022) Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients. Universal Journal of Mathematics and Applications 5 3 107–116.

IEEE

[1]H. Yin and Q. Wang, “Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients”, Univ. J. Math. Appl., vol. 5, no. 3, pp. 107–116, Sept. 2022, doi: 10.32323/ujma.1105072.

ISNAD

Yin, Hefan - Wang, Qi. “Numerical Stability of Runge-Kutta Methods for Differential Equations With Piecewise Constant Arguments With Matrix Coefficients”. Universal Journal of Mathematics and Applications 5/3 (September 1, 2022): 107-116. https://doi.org/10.32323/ujma.1105072.

JAMA

1.Yin H, Wang Q. Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients. Univ. J. Math. Appl. 2022;5:107–116.

MLA

Yin, Hefan, and Qi Wang. “Numerical Stability of Runge-Kutta Methods for Differential Equations With Piecewise Constant Arguments With Matrix Coefficients”. Universal Journal of Mathematics and Applications, vol. 5, no. 3, Sept. 2022, pp. 107-16, doi:10.32323/ujma.1105072.

Vancouver

1.Hefan Yin, Qi Wang. Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients. Univ. J. Math. Appl. 2022 Sep. 1;5(3):107-16. doi:10.32323/ujma.1105072