Document Zbl 1360.65200

The numerical asymptotically stability of a linear differential equation with piecewise constant arguments of mixed type. (English) Zbl 1360.65200

Acta Appl. Math. 146, No. 1, 145-161 (2016).

This work concerns the numerical stability of certain differential equations with piecewise constant arguments. A closed form expression to the solution for this problem is presented and its asymptotic stability is analysed. By applying Runge-Kutta methods to the differential equation, conditions under which the numerical solution is asymptotically stable are presented. The problem of when the analytical stability region is included in the numerical stability region is characterized. Results of numerical experiments validating the theoretical findings are reported.

Reviewer: Kanakadurga Sivakumar (Chennai)

Cited in 1 Document

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
65L07	Numerical investigation of stability of solutions to ordinary differential equations
65L20	Stability and convergence of numerical methods for ordinary differential equations

Keywords:

piecewise constant arguments; Runge-Kutta methods; \(\theta\)-methods; stability

Software:

RODAS

Cite Review PDF

Full Text: DOI

References:

[1]	Akhmet, M.U.: Stability of differential equations with piecewise constant arguments of generalized type. Nonlinear Anal. 68, 794-803 (2008) · Zbl 1173.34042 · doi:10.1016/j.na.2006.11.037
[2]	Chiu, K.S.: Stability of oscillatory solutions of differential equations with a general piecewise constant argument. Electron. J. Qual. Theory Differ. Equ. 88, 1-15 (2011) · Zbl 1340.34240 · doi:10.14232/ejqtde.2011.1.88
[3]	Muroya, Y.: New contractivity condition in a population model with piecewise constant arguments. J. Math. Anal. Appl. 346, 65-81 (2008) · Zbl 1161.34048 · doi:10.1016/j.jmaa.2008.05.025
[4]	Fu, X.L., Li, X.D.: Oscillation of higher order impulsive differential equations of mixed type with constant argument at fixed time. Math. Comput. Model. 48, 776-786 (2008) · Zbl 1156.34318 · doi:10.1016/j.mcm.2007.11.006
[5]	Wang, G.Q.: Periodic solutions of a neutral differential equation with piecewise constant arguments. J. Math. Anal. Appl. 326, 736-747 (2007) · Zbl 1113.34053 · doi:10.1016/j.jmaa.2006.02.093
[6]	Bainov, D., Kostadinov, T., Petrov, V.: Oscillatory and asymptotic properties of nonlinear first order neutral differential equations with piecewise constant argument. J. Math. Anal. Appl. 194, 612-639 (1995) · Zbl 0842.34070 · doi:10.1006/jmaa.1995.1321
[7]	Wiener, J.; Lakshmikantham, V. (ed.), Differential equations with piecewise constant delays, 547-580 (1983), New York
[8]	Wiener, J., Pointwise initial-value problems for functional differential equations, No. 92, 571-580 (1984) · Zbl 0552.34061
[9]	Cooke, K.L., Wiener, J.: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99, 256-297 (1984) · Zbl 0557.34059 · doi:10.1016/0022-247X(84)90248-8
[10]	Shah, S.M., Wiener, J.: Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci. 6, 671-703 (1983) · Zbl 0534.34067 · doi:10.1155/S0161171283000599
[11]	Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993) · Zbl 0874.34054 · doi:10.1142/1860
[12]	Wiener, J., Aftabizadeh, A.R.: Differential equations alternately of retarded and advanced type. J. Math. Anal. Appl. 129, 243-255 (1988) · Zbl 0671.34063 · doi:10.1016/0022-247X(88)90246-6
[13]	Alwan, M.S., Liu, X.Z., Xie, W.C.: Comparison principle and stability of differential equations with piecewise constant arguments. J. Franklin Inst. 350, 211-230 (2013) · Zbl 1278.93135 · doi:10.1016/j.jfranklin.2012.08.016
[14]	Györi, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford (1991) · Zbl 0780.34048
[15]	Liu, M.Z., Song, M.H., Yang, Z.W.: Stability of Runge-Kutta methods in the numerical solution of equation u′(t)=au(t)+a0u([t])\(u'(t)=au(t)+a_0u([t])\). J. Comput. Appl. Math. 166, 361-370 (2004) · Zbl 1052.65070 · doi:10.1016/j.cam.2003.04.002
[16]	Yang, Z.W., Liu, M.Z., Song, M.H.: Stability of Runge-Kutta methods in the numerical solution of equation u′(t)=au(t)+a0u([t])+a1u([t−1])\(u'(t)=au(t)+a_0u([t])+a_1u([t-1])\). Appl. Math. Comput. 162, 37-50 (2005) · Zbl 1063.65070
[17]	Liu, M.Z., Gao, J.F., Yang, Z.W.: Oscillation analysis of numerical solution in the θ \(\theta \)-methods for equation x′(t)+ax(t)+a1x([t−1])=\(0x'(t)+ax(t)+a_1x([t-1])=0\). Appl. Math. Comput. 186, 566-578 (2007) · Zbl 1118.65080
[18]	Liu, M.Z., Gao, J.F., Yang, Z.W.: Preservation of oscillations of the Runge-Kutta method for equation x′(t)+ax(t)+a1x([t−1])=\(0x'(t)+ax(t)+a_1x([t-1])=0\). Comput. Math. Appl. 58, 1113-1125 (2009) · Zbl 1189.65143 · doi:10.1016/j.camwa.2009.07.030
[19]	Lv, W.J., Yang, Z.W., Liu, M.Z.: Stability of the Euler-Maclaurin methods for neutral differential equations with piecewise continuous arguments. Appl. Math. Comput. 186, 1480-1487 (2007) · Zbl 1175.65088
[20]	Yang, Z.W., Liu, M.Z., Nieto, J.J.: Runge-Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. J. Comput. Appl. Math. 233, 990-1004 (2009) · Zbl 1180.65094 · doi:10.1016/j.cam.2009.08.105
[21]	Wang, W.S., Li, S.F.: Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay. Appl. Math. Lett. 21, 983-991 (2008) · Zbl 1152.65449 · doi:10.1016/j.aml.2007.10.014
[22]	Song, M.H., Liu, X.: The improved linear multistep methods for differential equations with piecewise continuous arguments. Appl. Math. Comput. 217, 4002-4009 (2010) · Zbl 1206.65184
[23]	Liang, H., Liu, M.Z., Yang, Z.W.: Stability analysis of Runge-Kutta methods for systems u′(t)=Lu(t)+Mu([t])\(u'(t)=Lu(t)+Mu([t])\). Appl. Math. Comput. 228, 463-476 (2014) · Zbl 1364.65144
[24]	Liang, H., Liu, M.Z., Lv, W.J.: Stability of θ \(\theta \)-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments. Appl. Math. Lett. 23, 198-206 (2010) · Zbl 1209.65091 · doi:10.1016/j.aml.2009.09.012
[25]	Liang, H., Shi, D.Y., Lv, W.J.: Convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument. Appl. Math. Comput. 217, 854-860 (2010) · Zbl 1204.65110
[26]	Wang, Q., Zhu, Q.Y.: Stability analysis of Runge-Kutta methods for differential equations with piecewise continuous arguments of mixed type. Int. J. Comput. Math. 88, 1052-1066 (2011) · Zbl 1220.65105 · doi:10.1080/00207160.2010.492425
[27]	Miller, J.J.H.: On the location of zeros of certion classes of polynomial with applications to numerical analysis. J. Inst. Math. Appl. 8, 397-406 (1971) · Zbl 0232.65070 · doi:10.1093/imamat/8.3.397
[28]	Song, M.H., Yang, Z.W., Liu, M.Z.: Stability of θ \(\theta \)-methods for advanced differential equations with piecewise continuous arguments. Comput. Math. Appl. 49, 1295-1301 (2005) · Zbl 1082.65078 · doi:10.1016/j.camwa.2005.02.002
[29]	Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. Wiley, New York (1987) · Zbl 0616.65072
[30]	Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993) · Zbl 0789.65048
[31]	Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems. Springer, Berlin (1996) · Zbl 0859.65067 · doi:10.1007/978-3-642-05221-7
[32]	Iserles, A., Nørsett, S.P.: Order stars and rational approximations to exp(z). Appl. Numer. Math. 5, 63-70 (1989) · Zbl 0674.65043 · doi:10.1016/0168-9274(89)90024-X

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.