scholarly journals Some Stability and Convergence of Additive Runge-Kutta Methods for Delay Differential Equations with Many Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Haiyan Yuan ◽  
Jingjun Zhao ◽  
Yang Xu
Keyword(s):  
Differential Equation ◽  
Numerical Solution ◽  
Convergence Analysis ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stability And Convergence ◽  
Runge Kutta ◽  
Lagrangian Interpolation ◽  
The Stability ◽  
Delay Differential

This paper is devoted to the stability and convergence analysis of the additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of a delay differential equation with many delays. GDN stability and D-Convergence are introduced and proved. It is shown that strongly algebraically stability gives D-Convergence DA, DAS, and ASI stability give GDN stability. Some examples are given in the end of this paper which confirms our results.

scholarly journals Nonlinear Stability and D-Convergence of Additive Runge-Kutta Methods for Multidelay-Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Haiyan Yuan ◽  
Jingjun Zhao ◽  
Yang Xu
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Convergence Analysis ◽  
Nonlinear Stability ◽  
Stability And Convergence ◽  
Numerical Example ◽  
Runge Kutta ◽  
Lagrangian Interpolation ◽  
The Stability ◽  
Theoretical Results

This paper is devoted to the stability and convergence analysis of the Additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of multidelay-integro-differential equations (MDIDEs). GDN-stability and D-convergence are introduced and proved. It is shown that strongly algebraically stability gives D-convergence, DA- DAS- and ASI-stability give GDN-stability. A numerical example is given to illustrate the theoretical results.

scholarly journals Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Neutral Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Haiyan Yuan ◽  
Cheng Song ◽  
Peichen Wang
Keyword(s):  
Differential Equations ◽  
Nonlinear Stability ◽  
Delay Differential Equations ◽  
Stability And Convergence ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
Restricted Type ◽  
Runge Kutta ◽  
The Stability ◽  
Delay Differential

This paper is devoted to the stability and convergence analysis of the two-step Runge-Kutta (TSRK) methods with the Lagrange interpolation of the numerical solution for nonlinear neutral delay differential equations. Nonlinear stability and D-convergence are introduced and proved. We discuss theGR(l)-stability,GAR(l)-stability, and the weakGAR(l)-stability on the basis of(k,l)-algebraically stable of the TSRK methods; we also discuss the D-convergence properties of TSRK methods with a restricted type of interpolation procedure.

scholarly journals Stability Threshold for Scalar Linear Periodic Delay Differential Equations

2016 ◽  
Vol 59 (4) ◽  
pp. 849-857
Author(s):  
Kyeongah Nah ◽  
Gergely Röst
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Trivial Solution ◽  
Delay Differential Equations ◽  
Stability Threshold ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Linear Scalar

AbstractWe prove that for the linear scalar delay differential equationwith non-negative periodic coefficients of period P > 0, the stability threshold for the trivial solution is , assuming that b(t+1)−a(t) does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that, in general, r = 0 is not a stability threshold.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

scholarly journals Non-Spiking Laser Controlled by a Delayed Feedback

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2069
Author(s):  
Anton V. Kovalev ◽  
Evgeny A. Viktorov ◽  
Thomas Erneux
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Optical Feedback ◽  
Laser Pulses ◽  
Equation Model ◽  
Rate Equations ◽  
Differential Equation Model ◽  
The Stability ◽  
Delay Differential ◽  
Intensity Laser

In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

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