Dynamic Behavior of Euler-Maclaurin Methods for Differential Equations with Piecewise Constant Arguments of Advanced and Retarded Type

Year 2021, Volume: 4 Issue: 3, 165 - 179, 30.09.2021

Hefan Yin , Qi Wang

https://doi.org/10.33401/fujma.906230

Abstract

The paper deals with three dynamic properties of the numerical solution for differential equations with piecewise constant arguments of advanced and retarded type: oscillation, stability and convergence. The Euler-Maclaurin methods are used to discretize the equations. According to the characteristic theory of the difference equation, the oscillation and stability conditions of the numerical solution are obtained. It is proved that the convergence order of numerical method is 2n+2. Furthermore, the relationship between stability and oscillation is discussed for analytic solution and numerical solution, respectively. Finally, several numerical examples confirm the corresponding conclusions.

Keywords

Euler-Maclaurin methods, numerical solution, oscillation, stability, convergence

Supporting Institution

the Natural Science Foundation of Guangdong Province

Project Number

2017A030313031

Thanks

Thanks for the Natural Science Foundation of Guangdong Province to support this study.

References

• [1] A. Konuralp, S. Oner, Numerical solutions based on a collocation method combined with Euler polynomials for linear fractional differential equations with delay, Int. J. Nonlin. Sci. Num., 21(6) (2020), 539-547.
• [2] K. S. Brajesh, A. Saloni, A new approximation of conformable time fractional partial differential equations with proportional delay, Appl. Numer. Math., 157 (2020), 419-433.
• [3] G. P. Wei, J. H. Shen, Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients, Math. Comput. Model., 44(11-12) (2018), 1089-1096.
• [4] G. L. Zhang, M. H. Song, Impulsive continuous Runge-Kutta methods for impulsive delay differential equations, Appl. Math. Comput., 341 (2019), 160-173.
• [5] C. J. Zhang, C. Li, J. Y. Jiang, Extended block boundary value methods for neural equations with piecewise constant argument, Appl. Numer. Math., 150 (2020), 182-193.
• [6] K. S. Chiu, T. X. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr., 292 (2019), 2153-2164.
• [7] K. S. Chiu, J. C. Jeng, Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Math. Nachr., 288(10) (2015), 1085-1097.
• [8] M. Esmailzadeh, H. S. Najafi, H. Aminikhah, A numerical scheme for diffusion-convection equation with piecewise constant argument, Comput. Methods Differ. Equ., 8(3) (2020), 573-584.
• [9] X. Y. Li, H. X. Li, B. Y. Wu, Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments, Appl. Math. Comput., 349 (2019), 304-313.
• [10] F. Karakoc, Asymptotic behaviour of a population model with piecewise constant argument, Appl. Math. Lett., 70 (2017), 7-13.
• [11] T. H. Yu, D. Q. Cao, Stability analysis of impulsive neural networks with piecewise constant arguments, Neural. Process. Lett., 47(1) (2018), 153-165.
• [12] K. S. Chiu, M. Pinto, J. C. Jeng, Existence and global convergence of periodic solutions in the current neural network with a general piecewise alternately advanced and retarded argument, Acta Appl. Math., 133 (2014), 133-152.
• [13] S. Kartal, F. Gurcan, Global behaviour of a predator-prey like model with piecewise constant arguments, J. Biol. Dynam., 9(1) (2015), 159-171.
• [14] F. Bozkurt, A. Yousef, T. Abdeljawad, Analysis of the outbreak of the novel coronavirus COVID-19 dynamic model with control mechanisms, Results in Physics, 19 (2020), 103586.
• [15] 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.
• [16] Y. L. Lu, M. H. Song, M. Z. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math., 317 (2017), 55-71.
• [17] W. S. Wang, Stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations, Appl. Math. Comput., 219(9) (2013), 4590-4600.
• [18] Q. Wang, J. Y. Yao, Numerical stability and oscillation of a kind of functional differential equations, J. Liaocheng Univ. (Nat. Sci.), 33(2) (2020), 18-27.
• [19] 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., 288 (2014), 463-476.
• [20] S. M. Shah, J. Wiener, Advanced differential equations with piecewise constant argument deviations, Int. J. Math. Math. Sci., 6 (4), 671-703.
• [21] W. J. Lv, Z. W. Yang, M. Z. Liu, Stability of the Euler-Maclaurin methods for neutral differential equations with piecewise continuous arguments, Appl. Math. Comput., 106 (2007), 1480-1487.
• [22] J. Stoer, R. Bulirsh (editors), Introduction to Numerical Analysis, New York, Springer, 1993, pp. 156-160.