IJISA Vol. 3, No. 4, 8 Jun. 2011

Cover page and Table of Contents: PDF (size: 108KB)

Full Text (PDF, 108KB), PP.49-55

Views: 0 Downloads: 0

Runge-Kutta methods, numerical solution, piecewise constant arguments, oscillation

The purpose of this paper is to study the numerical oscillations of Runge-Kutta methods for the solution of alternately advanced and retarded differential equations with piecewise constant arguments. The conditions of oscillations for the Runge-Kutta methods are obtained. It is proven that the Runge-Kutta methods preserve the oscillations of the analytic solution. In addition, the relationship between stability and oscillations are shown. Some numerical examples are given to confirm the theoretical results.

Qi Wang, FengLian Fu, "Numerical Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Alternately Advanced and Retarded Type", International Journal of Intelligent Systems and Applications(IJISA), vol.3, no.4, pp.49-55, 2011. DOI:10.5815/ijisa.2011.04.07

[1]H. Khatibzadeh, “An oscillation criterion for a delay difference equation,” Comput. Math. Appl., vol. 57, pp. 37–41, 2009.

[2]Q.L. Li, Z.G. Zhang, F. Guo, Z.Y. Liu, and H.Y. Liang, “Oscillatory criteria for Third-Order difference equation with impulses,” J. Comput. Appl. Math., vol. 225, pp. 80–86, 2009.

[3]B.G. Jia, L. Erbe, and A. Peterson, “New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales,” Comput. Math. Appl. vol.56, pp. 2744-2756, 2008.

[4]B.G. Jia, L. Erbe, and A. Peterson, “Oscillation of sublinear Emden-Fowler dynamic equations on time scales,” J. Difference Equations Appl. vol. l6, pp. 217-226, 2010.

[5]J.R. Graef, J.H. Shen, and I.P. Stavroulakis, “Oscillation of impulsive neutral delay differential equations,” J. Math. Anal. Appl. vol. 268, pp. 310-333, 2002.

[6]I. Kubiaczyk, S.H. Saker, “Oscillation and stability in nonlinear delay differential equations of population dynamics,” Math. Comput. Model., vol. 35, pp. 295-301, 2002.

[7]L.P. Gimenes, M. Federson, “Oscillation by impulses for a second-order delay differential equation, ” Comput. Math. Appl., vol. 52, pp. 819-828, 2006.

[8]A.Z. Weng, J.T. Sun, “Oscillation of second order delay differential equations,” Appl. Math. Comput., vol. 198, pp. 930-935, 2008.

[9]J.H. Shen, I.P. Stavroulakis, “Oscillatory and nonoscillatory delay equations with piecewise constant argument,” J. Math. Anal. Appl., vol. 248, pp. 385-401, 2000.

[10]Z.G. Luo, J.H. Shen, “New results on oscillation for delay differential equations with piecewise constant argument,” Comput. Math. Appl., vol. 45, pp. 1841-1848, 2003.

[11]Y.B. Wang, J.R. Yan, “Oscillation of a differential equation with fractional delay and piecewise constant arguments,” Comput. Math. Appl., vol. 52, pp. 1099-1106, 2006.

[12]X.L. Fu, X.D. Li, “Oscillation of higher order impulsive differential equations of mixed type with constant argument at fixed time,” Math. Comput. Model., vol. 48, pp. 776-786, 2008.

[13]S. Busenberg, K. Cooke, Vertically Transmitted Diseases, Models and Dynamics, in: Biomathematics, Berlin: Springer, 1993.

[14]T. Kupper, R. Yuang, “On quasi-periodic solutions of differential equations with piecewise constant argument,” J. Math. Anal. Appl., vol. 267, pp. 173-193, 2002.

[15]G. Papaschinopoulos, “Linearization near the integral manifold for a system of differential equations with piecewise constant argument,” J. Math. Anal. Appl., vol. 215, pp. 317-333, 1997.

[16]A. Alonso, J.L. Hong, “Ergodic type solutions of differential equations with piecewise constant arguments,” Int. J. Math. Math. Sci., vol. 28, pp. 609-619, 2001.

[17]Y. Muroya, “Persistence, contractivity and global stability in logistic equations with piecewise constant delays,” J. Math. Anal. Appl., vol. 270, pp. 602-635, 2002.

[18]R. Yuan, “On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument,” J. Math. Anal. Appl., vol. 303, pp. 103-118, 2005.

[19]F. Gurcan, F. Bozkurt, “Global stability in a population model with piecewise constant arguments,” J. Math. Anal. Appl., vol. 360, pp. 334-342, 2009.

[20]J. Wiener, Generalized Solutions of Functional Differential Equations. Singapore: World Scientific, 1993.

[21]M.Z. Liu, M.H. Song, and Z.W. Yang, “Stability of Runge-Kutta methods in the numerical solution of equation ,” J. Comput. Appl. Math., vol. 166, pp. 361-370, 2004.

[22]Z.W. Yang, M.Z. Liu, and M.H. Song, “Stability of Runge-Kutta methods in the numerical solution of equation ,” Appl. Math. Comput., vol. 162, pp. 37-50, 2005.

[23]M.Z. Liu, S.F. Ma, and Z.W. Yang, “Stability analysis of Runge-Kutta methods for unbounded retarded differential equations with piecewise continuous arguments,” Appl. Math. Comput., vol. 191, pp. 57-66, 2007.

[24]W.J. Lv, Z.W. Yang, and M.Z. Liu, “Stability of Runge-Kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments,” Comput. Math. Appl., vol. 54, pp. 326-335, 2007.

[25]M.Z. Liu, J.F. Gao, and Z.W. Yang, “Oscillation analysis of numerical solution in the -methods for equation ,” Appl. Math. Comput., vol. 186, pp. 566-578, 2007.

[26]M.Z. Liu, J.F. Gao, and Z.W. Yang, “Preservation of oscillations of the Runge-Kutta method for equation ,” Comput. Math. Appl., vol. 58, pp. 1113-1125, 2009.

[27]A.R. Aftabizadeh, J. Wiener, “Oscillatory and periodic solutions of an equation alternately of retarded and advanced type,” Appl. Anal., vol. 23, pp. 219-231, 1986.