## Preservation of oscillations of the Runge-Kutta method for equation $$x'(t)+ax(t)+a_1x([t - 1])=0$$.(English)Zbl 1189.65143

Summary: The paper deals with the preservation of oscillations of the Runge-Kutta method for equation $$x'(t)+ax(t)+a_{1}x([t - 1])=0$$. It is proved that oscillations of the analytic solution are preserved by the Runge-Kutta method. Special interpolation functions of the numerical solutions are given. It turns out that zeros of the interpolation function of the numerical solution converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge-Kutta method. Some numerical experiments are presented.

### MSC:

 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L03 Numerical methods for functional-differential equations
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### References:

 [1] Akhmet, M.U., On the reduction principle for differential equations with piecewise constant argument of generalized type, J. math. anal. appl., 336, 646-663, (2007) · Zbl 1134.34048 [2] Cabada, A.; Ferreiro, J.B.; Nieto, J.J., Green’s function and comparison principles for first order periodic differential equations with piecewise constant arguments, J. math. anal. appl., 291, 690-697, (2004) · Zbl 1057.34089 [3] Nieto, J.J.; Rodriguez-Lopez, R., Green’s function for second-order periodic boundary value problems with piecewise constant arguments, J. math. anal. appl., 304, 33-57, (2005) · Zbl 1078.34046 [4] 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 [5] Akhmet, M.U., Asymptotic behavior of solutions of differential equations with piecewise constant arguments, Appl. math. lett., 21, 951-956, (2008) · Zbl 1152.34348 [6] Li, H.; Muroya, Y.; Yuan, R., A sufficient condition for the global asymptotic stability of a class of logistic equations with piecewise constant delay, Nonlinear anal. RWA, 10, 244-253, (2009) · Zbl 1154.34383 [7] Györi, I.; Hartung, F., On numerical approximation using differential equations with piecewise-constant arguments, Period. math. hungar., 56, 55-69, (2008) · Zbl 1164.34038 [8] Akhmet, M.U., Stability of differential equations with piecewise constant arguments of generalized type, Nonlinear anal., 68, 794-803, (2008) · Zbl 1173.34042 [9] Wang, Y.B.; Yan, J.R., Oscillation of a differential equation with fractional delay and piecewise constant arguments, Comput. math. appl., 52, 1099-1106, (2006) · Zbl 1132.34047 [10] Luo, Z.G.; Shen, J.H., New results on oscillation for delay differential equations with piecewise constant argument, Comput. math. appl., 45, 1841-1848, (2003) · Zbl 1065.34061 [11] Shen, J.H., Oscillatory and nonoscillatory delay equations with piecewise constant argument, J. math. anal. appl., 248, 385-401, (2000) · Zbl 0966.34063 [12] Liu, M.Z.; Song, M.H.; Yang, Z.W., Stability of runge – kutta methods in the numerical solution of equation $$u^\prime(t) = a u(t) + a_0 u([t])$$, J. comput. appl. math., 166, 361-370, (2004) · Zbl 1052.65070 [13] Song, M.H.; Yang, Z.W.; Liu, M.Z., Stability of $$\theta$$-methods for advanced differential equations with piecewise continuous arguments, Comput. math. appl., 49, 1295-1301, (2005) · Zbl 1082.65078 [14] Yang, Z.W.; Liu, M.Z.; Song, M.H., Stability of runge – kutta methods in the numerical solution of equation $$u^\prime(t) = a u(t) + a_0 u([t]) + a_1 u([t - 1])$$, Appl. math. comput., 162, 37-50, (2005) · Zbl 1063.65070 [15] Liu, M.Z.; Gao, J.F.; Yang, Z.W., Oscillation analysis of numerical solution in the $$\theta$$-methods for equation $$x^\prime(t) + a x(t) + a_1 x([t - 1]) = 0$$, Appl. math. comput., 186, 566-578, (2007) · Zbl 1118.65080 [16] Figueroa-Sestelo, R.; Lpez-Pouso, R., Discontinuous first-order functional boundary value problems, Nonlinear anal., 69, 2142-2149, (2008) · Zbl 1210.34088 [17] Jiang, D.; Nieto, J.J.; Zuo, W., On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations, J. math. anal. appl., 289, 691-699, (2004) · Zbl 1134.34322 [18] Nieto, J.J.; Rodriguez-Lopez, R., Monotone method for first-order functional differential equations, Comput. math. appl., 5, 471-484, (2006) · Zbl 1140.34406 [19] Nieto, J.J.; Rodriguez-Lopez, R., Boundary value problems for a class of impulsive functional equations, Comput. math. appl., 5, 2715-2731, (2008) · Zbl 1142.34362 [20] Hadd, S., Singular functional differential equations of neutral type in Banach spaces, J. funct. anal., 254, 2069-2091, (2008) · Zbl 1139.47032 [21] Wiener, J., Generalized solutions of functional differential equations, (1993), World Scientific Singapore · Zbl 0874.34054 [22] Aftabizadeh, A.R.; Wiener, J.; Xu, J.M., Oscillation and periodic solutions of delay differential equations with piecewise constant argument, Proc. amer. math. soc., 99, 673-679, (1987) · Zbl 0631.34078 [23] Györi, I.; Ladas, G., Oscillation theory of delay equations: with applications, (1991), Clarendon Press Oxford · Zbl 0780.34048 [24] Liu, M.Z.; Ma, S.F.; Yang, Z.W., Stability analysis of runge – kutta methods for unbounded retarded differential equations with piecewise continuous arguments, Appl. math. comput., 191, 57-66, (2007) · Zbl 1193.65122 [25] Lambert, J.D., Numerical methods for ordinary differential systems, (1991), John Wiley · Zbl 0745.65049 [26] Bellen, A.; Zennaro, M., Numerical methods for delay differential equations, (2003), Clarendon Press Oxford, pp. 110-118
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