scholarly journals On stability sets for numerical discretizations of neutral delay differential equations

2015 ◽  
Vol 63 (1) ◽  
pp. 89-100
Author(s):  
Jan Čermák ◽  
Jana Dražková
Keyword(s):  
Differential Equation ◽  
Positive Real ◽  
Neutral Delay ◽  
Recent Developments ◽  
Constant Coefficients ◽  
The Stability ◽  
Delay Differential ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Stability Sets

Abstract The paper discusses the -method discretization of the neutral delay differential equation y'(t) = ay (t) + by (t - τ) + cy' (t - τ), t > 0, where a, b, c are real constant coefficients and is a positive real lag. Using recent developments on stability of appropriate delay difference equations we give a complete description of stability sets for this discretization. Some of their properties and related comparisons with the stability set for the underlying neutral differential equation are discussed as well.

Stability of Human Balance With Reflex Delays Using Galerkin Approximations

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Zaid Ahsan ◽  
Thomas K. Uchida ◽  
Akash Subudhi ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equation ◽  
Galerkin Approximation ◽  
The Elderly ◽  
Double Pendulum ◽  
Neutral Delay ◽  
Stability Margins ◽  
Human Balance ◽  
The Stability ◽  
The Galerkin Method ◽  
Delay Differential

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation (PDE) constrained by a boundary condition and then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. We validate our formulation by deriving analytical expressions for the stability margins of the double-pendulum human stance model. Numerical examples demonstrate that proportional–derivative–acceleration (PDA) feedback generally, but not always, results in larger stability margins than proportional–derivative (PD) feedback in the presence of reflex delays.

scholarly journals Necessary and sufficient conditions for oscillation of first order neutral delay difference equations

2015 ◽  
Vol 3 (2) ◽  
pp. 61
Author(s):  
A. Murgesan ◽  
P. Sowmiya
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
First Order ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference

<p>In this paper, we obtained some necessary and sufficient conditions for oscillation of all the solutions of the first order neutral delay difference equation with constant coefficients of the form <br />\begin{equation*} \quad \quad \quad \quad \Delta[x(n)-px(n-\tau)]+qx(n-\sigma)=0, \quad \quad n\geq n_0 \quad \quad \quad \quad \quad \quad {(*)} \end{equation*}<br />by constructing several suitable auxiliary functions. Some examples are also given to illustrate our results.</p>

Two types of stability conditions for linear delay difference equations

2015 ◽  
Vol 9 (1) ◽  
pp. 120-138 ◽  
Author(s):  
Jan Cermák ◽  
Jiří Jánský ◽  
Petr Tomásek
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Linear Difference Equation ◽  
Stability Conditions ◽  
Linear Delay ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Asymptotic Stability Conditions

The paper discusses asymptotic stability conditions for a four-parameter linear difference equation appearing in the process of discretization of a delay differential equation. We present two types of conditions, which are necessary and sufficient for asymptotic stability of the studied equation. A relationship between both the types of conditions is established and some of their consequences are discussed.

scholarly journals Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
K. C. Panda ◽  
R. N. Rath ◽  
S. K. Rath
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Oscillation And Nonoscillation

In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation yt−∑j=1kpjtyrjt′+qtGygt−utHyht=ft, where pj and rj for each j and q,u,G,H,g,h, and f are all continuous functions and q≥0,u≥0,ht<t,gt<t, and rjt<t for each j. Further, each rjt, gt, and ht⟶∞ as t⟶∞. This paper improves and generalizes some known results.

Preliminary Study on Stick-Slip in Drillstring With Analytical Model Expressed With Neutral Delay Differential Equation

2014 ◽  
Author(s):  
Tomoya Inoue ◽  
Tokihiro Katsui ◽  
Chang-Kyu Rheem ◽  
Zengo Yoshida ◽  
Miki Y. Matsuo
Keyword(s):  
Differential Equation ◽  
Analytical Model ◽  
Delay Differential Equation ◽  
Drill Bit ◽  
Neutral Delay ◽  
Stick Slip ◽  
Scientific Drilling ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Slip Phenomenon

Stick-slip is a major problem in offshore drilling because it may cause damage to the drill bit as well as crushing or grinding the sediment layer, which is crucial problem in scientific drilling because the purpose of the scientific drilling is to recover core samples from the layers. To mitigate stick-slip, first of all it is necessary to establish a model of the torsional motion of the drill bit and express the stick-slip phenomenon. Toward this end, the present study proposes a model of torsional waves propagating in a drillstring. An analytical model is developed and used to derive a neutral delay differential equation (NDDE), a special type of equation that requires time history, and an analytical model of stick-slip is derived for friction models between the drill bit and the layer as well as the rotation speed applied to the uppermost part of the drill string. In this study, the stick-slip model is numerically analyzed for several conditions and a time series of the bit motions is obtained. Based on the analytical results, the appearance of stick-slip and its severity are discussed. A small-scale model experiment was conducted in a water tank to observe the stick-slip phenomenon, and the result is discussed with numerical analysis. In addition, utilizing surface drilling data acquired from the actual drilling operations of the scientific drillship Chikyu, occurrence of stick-slip phenomenon is discussed.

scholarly journals Oscillation Criteria for Second Order Neutral Differential Equations

1996 ◽  
Vol 48 (4) ◽  
pp. 871-886 ◽  
Author(s):  
Horng-Jaan Li ◽  
Wei-Ling Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Image Position ◽  
Delay Differential

AbstractSome oscillation criteria are given for the second order neutral delay differential equationwhere τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

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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