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Stability of analytic and numerical solutions for differential equations with piecewise continuous arguments. (English) Zbl 1244.34097
Summary: The asymptotic stability of the analytic and numerical solutions for differential equations with piecewise continuous arguments is investigated by using Lyapunov’s method. In particular, linear equations with variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical solutions of the $\theta$-methods are obtained. Some examples are illustrated.

34K20Stability theory of functional-differential equations
34K28Numerical approximation of solutions of functional-differential equations
34K06Linear functional-differential equations
Full Text: DOI
[1] J. Wiener, “Differential equations with piecewise constant delays,” in Trends in the Theory and Practice of Nonlinear Differential Equations, V. Lakshmikantham, Ed., pp. 547-552, Marcel Dekker, New York, NY, USA, 1983. · Zbl 0531.34059
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[7] M. H. Song, Z. W. Yang, and M. Z. Liu, “Stability of \theta -methods for advanced differential equations with piecewise continuous arguments,” Computers & Mathematics with Applications, vol. 49, no. 9-10, pp. 1295-1301, 2005. · Zbl 1082.65078 · doi:10.1016/j.camwa.2005.02.002
[8] Z. Yang, M. Liu, and M. Song, “Stability of Runge-Kutta methods in the numerical solution of equation u$^{\prime}$(t) = au(t) + a0u([t]) + a1u([t - 1]),” Applied Mathematics and Computation, vol. 162, no. 1, pp. 37-50, 2005. · Zbl 1063.65070 · doi:10.1016/j.amc.2003.12.081
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