Stability and Oscillation of θ-methods for Differential Equation with Piecewise Constant Arguments
Keywords:
θ-methods, Stability, Oscillation, Piecewise Constant Arguments
Abstract
This paper studies the numerical properties of θ-methods for the alternately advanced and retarded differential equation u′(t) = au(t)+bu(2[(t+1)/2]). Using two classes of θ-methods, namely the linear θ-method and the one-leg θ-method, the stability regions of numerical methods are determined, and the conditions of oscillation for the θ-methods are derived. Moreover, we give the conditions under which the numerical stability regions contain the analytical stability regions. It is shown that the θ-methods preserve the oscillation of the analytic solution. In addition, the relationships between stability and oscillation are presented. Several numerical examples are given.
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References
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[2] Chiu, K.S. (2021) Global exponential stability of bidirectional associative memory neural networks model with piecewise alternately advanced and retarded argument. Comput. Appl. Math., 40(8).
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[9] Liang, H., Liu, M.Z., and Yang, Z.W. (2014) Stability analysis of Runge-Kutta methods for systems u′ (t) = Lu(t) + Mu([t]). Appl. Math. Comput., 228: 463-476.
[10] Liu, M.Z., Gao, J.F., and Yang, Z.W. (2007) Oscillation analysis of numerical solution in the θ-methods for equation x′(t) + ax(t) + a1x([t − 1]) = 0. Appl. Math. Comput., 186(1): 566-578.
[11] Liu, M.Z., Gao, J.F., and Yang, Z.W. (2009). Preservation of oscillations of the Runge-Kutta method for equation x′(t) + ax(t) + a1x([t − 1]) = 0. Comput. Math. Appl., 58(6): 1113-1125.
[12] Liu, P.Z. and Gopalsamy, K. (1999). Global stability and chaos in a population model with piecewise constant arguments. Appl. Math. Comput., 101(1): 63-88.
[13] Muroya, Y. (2008). New contractivity condition in a population model with piecewise constant arguments. J. Math. Anal. Appl., 346(1): 65-81.
[14] Shah, S.M. and Wiener, J. (1983). Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci., 6(4): 671-703.
[15] Song, M.H. and Liu, M.Z. (2012). Numerical stability and oscillation of the Runge-Kutta methods for the differential equations with piecewise continuous arguments alternately of retarded and advanced type. J. Inequal. Appl., 2012: 1-13.
[16] Song, M.H., Yang, Z.W., and Liu, M.Z. (2005). Stability of θ-methods for advanced differential equations with piecewise continuous arguments. Comput. Math. Appl., 49(9-10): 1295-1301.
[17] Wang, G.Q. (2007). Periodic solutions of a neutral differential equation with piecewise constant arguments. J. Math. Anal. Appl., 326(1): 736-747.
[18] Wang, Q., Zhu, Q.Y., and Liu, M.Z. (2011). Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type. J. Comput. Appl. Math., 235(5): 1542-1552.
[19] Wang, W.S. (2013). Stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations. Appl. Math. Comput., 219(9): 4590-4600.
[20] Wiener, J. and Aftabizadeh, A.R. (1988). Differential equations alternately of retarded and advanced type. J. Math. Anal. Appl., 129(1): 243-255.
[21] Wiener, J. and Cooke, K.L. (1989). Oscillations in systems of differential equations with piecewise constant argument. J. Math. Anal. Appl., 137(1): 221-239.
[22] Wiener, J. (1993). Generalized Solutions of Functional Differential Equations. World Scientific, Singapore.
[23] Yang, Z.W., Liu, M.Z., and Nieto, J.J. (2009). Runge-Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. Comput. Math. Appl., 233(4): 990-1004.
[24] Yin, H.F. and Wang, Q. (2021). Dynamic behavior of Euler-Maclaurin methods for differential equations with piecewise constant arguments of advanced and retarded type. Fund. J. Math. Appl., 4(3): 165-179.
[2] Chiu, K.S. (2021) Global exponential stability of bidirectional associative memory neural networks model with piecewise alternately advanced and retarded argument. Comput. Appl. Math., 40(8).
[3] Cooke, K.L. and Wiener, J. (1984). Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 99(1): 265-297.
[4] Cooke, K.L. and Wiener, J. (1987). An equation alternately of retarded and advanced type. Proc. Amer. Math. Soc., 99: 726-732.
[5] Dhama, S., Abbas, S., and Sakthivel, R. (2021). Stability and approximation of almost automorphic solutions on time scales for the stochastic Nicholson’s blowflies model . J. Integral Equ. Appl., 33(1): 31-51.
[6] Jayasree, K.N. and Deo, S.G. (1991). Variation of parameters formula for the equation of Cooke and Wiener. Proc. Amer. Math. Soc., 112(1): 75-80.
[7] Kocic, V.L. and Ladas, G. (1993). Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, Dordrecht.
[8] K¨upper, T. and Yuang, R. (2002). On quasi-periodic solutions of differential equations with piecewise constant argument. J. Math. Anal. Appl., 267(1): 173-193.
[9] Liang, H., Liu, M.Z., and Yang, Z.W. (2014) Stability analysis of Runge-Kutta methods for systems u′ (t) = Lu(t) + Mu([t]). Appl. Math. Comput., 228: 463-476.
[10] Liu, M.Z., Gao, J.F., and Yang, Z.W. (2007) Oscillation analysis of numerical solution in the θ-methods for equation x′(t) + ax(t) + a1x([t − 1]) = 0. Appl. Math. Comput., 186(1): 566-578.
[11] Liu, M.Z., Gao, J.F., and Yang, Z.W. (2009). Preservation of oscillations of the Runge-Kutta method for equation x′(t) + ax(t) + a1x([t − 1]) = 0. Comput. Math. Appl., 58(6): 1113-1125.
[12] Liu, P.Z. and Gopalsamy, K. (1999). Global stability and chaos in a population model with piecewise constant arguments. Appl. Math. Comput., 101(1): 63-88.
[13] Muroya, Y. (2008). New contractivity condition in a population model with piecewise constant arguments. J. Math. Anal. Appl., 346(1): 65-81.
[14] Shah, S.M. and Wiener, J. (1983). Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci., 6(4): 671-703.
[15] Song, M.H. and Liu, M.Z. (2012). Numerical stability and oscillation of the Runge-Kutta methods for the differential equations with piecewise continuous arguments alternately of retarded and advanced type. J. Inequal. Appl., 2012: 1-13.
[16] Song, M.H., Yang, Z.W., and Liu, M.Z. (2005). Stability of θ-methods for advanced differential equations with piecewise continuous arguments. Comput. Math. Appl., 49(9-10): 1295-1301.
[17] Wang, G.Q. (2007). Periodic solutions of a neutral differential equation with piecewise constant arguments. J. Math. Anal. Appl., 326(1): 736-747.
[18] Wang, Q., Zhu, Q.Y., and Liu, M.Z. (2011). Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type. J. Comput. Appl. Math., 235(5): 1542-1552.
[19] Wang, W.S. (2013). Stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations. Appl. Math. Comput., 219(9): 4590-4600.
[20] Wiener, J. and Aftabizadeh, A.R. (1988). Differential equations alternately of retarded and advanced type. J. Math. Anal. Appl., 129(1): 243-255.
[21] Wiener, J. and Cooke, K.L. (1989). Oscillations in systems of differential equations with piecewise constant argument. J. Math. Anal. Appl., 137(1): 221-239.
[22] Wiener, J. (1993). Generalized Solutions of Functional Differential Equations. World Scientific, Singapore.
[23] Yang, Z.W., Liu, M.Z., and Nieto, J.J. (2009). Runge-Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. Comput. Math. Appl., 233(4): 990-1004.
[24] Yin, H.F. and Wang, Q. (2021). Dynamic behavior of Euler-Maclaurin methods for differential equations with piecewise constant arguments of advanced and retarded type. Fund. J. Math. Appl., 4(3): 165-179.
Published
2022-01-17
How to Cite
Wang, Q., & Liu, X. (2022). Stability and Oscillation of θ-methods for Differential Equation with Piecewise Constant Arguments. Journal of Progressive Research in Mathematics, 19(1), 1-16. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2116
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