Methods For Delay Differential Equations Research Articles

Computing stability regions—Runge-Kutta methods for delay differential equations

Computing stability regions—Runge-Kutta methods for delay differential equations

The stability of a class of Runge-Kutta methods for delay differential equations

This paper deals with the stability analysis of Runge-Kutta type methods for delay differential equations. We focus on the subclass of collocation methods with abscissae in [0,1), and we prove that all of these methods violate an important stability condition related to the class of testproblems U′( t) = λU( t) + μU( t− τ) with λ,μ ∈ C ,Re λ < − ∣ μ ;∣ and τ > 0.

Stability analysis of numerical methods for delay differential equations

This paper deals with the stability analysis of step-by-step methods for the numerical solution of delay differential equations. We focus on the behaviour of such methods when they are applied to the linear testproblemU?(t)=?U(t)+μU(t??) with ?>0 and ?, μ complex. A general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods.

On the asymptotic stability of?-methods for delay differential equations

Stability regions of θ-methods for the linear delay differential test equations $$\begin{gathered} y'(t) = py(t) + qy(t - \tau ),t > 0, \hfill \\ y(t) = \varphi (t),t \in [ - \tau ,0], \hfill \\ \end{gathered}$$ where τ is a positive constant, are presented. In the case thatp andq are real constant coefficients, necessary and sufficient conditions on the stepsize for the stability of a θ-method are obtained. Furthermore, whenp andq are complex coefficients, sufficient conditions for the stability of the θ-methods are also given.

Stability of numerical methods for delay differential equations

Consider the following delay differential equation (DDE) y′=ƒ(t,y(t),y(t−τ(t))), t⩾t 0 ,with the initial condition y( t)= Φ for t⩽ t 0,where ƒ and Φ are such that (0.1), (0.2) has a unique solution y( t). The author gives sufficient conditions for the asymptotic stability of the equation (0.1) for which he introduces new definitions of numerical stability. The approach is reminiscent of that from the nonlinear, stiff ordinary differential equation (ODE) field. He investigates stability properties of the class of one-point collocation rules. In particular, the backward Euler method turns out to be stable with respect to all the given definitions.

P-stability properties of Runge-Kutta methods for delay differential equations

Recently the author and others have studied a class of Runge-Kutta methods for DDEs. These methods make use of certain constrained meshes, which allow to get optimal order results. In this paper we study their asymptotic stability properties, using the test equation $$\begin{gathered} y'(t) = ay(t) + by(t - \tau ),t > 0 \hfill \\ y(t) = g(t)for - \tau \leqq t \leqq 0 \hfill \\ \end{gathered} $$ wherea,b∈ℂ, σ>0, andg(t) is continuous and complex valued. It is known thaty(t)→0 ast→+∞ if |b|<-Re(a). In particular, we show that any Astable one-step collocation method for ODEs inherits the same property when it is applied to DDEs with a constrained mesh (i.e. it is P-stable).

Stability analysis of linear multistep methods for delay differential equations

Stability properties of linear multistep methods for delay differential equations with respect to the test equation urn:x-wiley:01611712:media:ijmm768965:ijmm768965-math-0001 0 &lt; λ &lt; 1, are investigated. It is known that the solution of this equation is bounded if and only if |a| &lt; −b and we examine whether this property is inherited by multistep methods with Lagrange interpolation and by parametrized Adams methods.

A one-step subregion method for delay differential equations

We study a one-step method for delay differential equations, which is equivalent to an implicit Runge-Kutta method. It approximates the solution in the whole interval with a piecewise polynomial of fixed degree n. For an appropiate choice of the mesh points, it provides uniform convergence 0(hn+1) and the superconvergence 0(h2n) at the nodes.

Stability properties of numerical methods for solving delay differential equations

Stability properties of numerical methods for delay differential equations are considered. Some suitable definitions for the stability of the numerical methods are included and Runge-Kutta type methods satisfying these properties are tested on a numerical example.

Fixed step discretisation methods for delay differential equations

The convergence of a fairly wide class of fixed step linear discretisation methods for delay differential equations with variable delay is studied. A global order convergence result which permits the case when discontinuities are present is given. This utilises a new formulation of the root condition for Dahlquist stability. Numerical results illustrating the theory are presented.

Stability of Multistep Methods for Delay Differential Equations

The absolute and relative stability of linear multistep methods for a finite step size is studied for delay differential equations. The differential equations are assumed linear and the delays a constant integer multiple of the step size. Computable conditions for stability are developed for scalar equations. Plots of the stability regions for several common multistep methods are included. For the integration methods considered, the stability regions for delay differential equations are significantly different from the stability regions for ordinary differential equations.

Stability of multistep methods for delay differential equations

The absolute and relative stability of linear multistep methods for a finite step size is studied for delay differential equations. The differential equations are assumed linear and the delays a constant integer multiple of the step size. Computable conditions for stability are developed for scalar equations. Plots of the stability regions for several common multistep methods are included. For the integration methods considered, the stability regions for delay differential equations are significantly different from the stability regions for ordinary differential equations.

Excitation and measurement of acoustic ion waves : Sessler, G.M. Journal of the Acoustical Society of America, 42, No 2, p 360 (August 1967)

Stability analysis of surge oscillations of two-point mooring system under state feedback control with time-delay is investigated. The two-point mooring system is harmonically excited and essentially represents a strongly nonlinear Duffing oscillator. In this paper, a frequency domain based method viz. incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary and higher order subharmonic responses which may be present in such system. The IHBC is then reformulated in a manner to treat two-point mooring system under state feedback control with time-delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses for primary and higher order subharmonic oscillations is obtained by Floquet’s theory using Hsu’ scheme; whereas the stability of controlled responses is obtained by applying semi-discretization method for delay differential equation. The study focussed on the controlling primary, higher order subharmonics and chaotic responses by considering appropriate feedback gains and delay by way of (i) appreciable reduction of primary, subharmonic responses, (ii) exclusion of all higher order subharmonics 2T, 3T, 5T and 9T (1/n subharmonics or period-n solutions), and (iii) reduction of the extent of domain of all instability phenomena represented by various type of bifurcation of solutions, jump phenomena, chaotic responses etc. In the study, negative velocity feedback is observed to be much effective than state feedback for better controlling of surge oscillation of two-point mooring system. Also, the effect of larger gain values is investigated by an extensive parametric study for vibration control with different delay values.