Methods For Delay Differential Equations Research Articles

Numerical methods for delay differential equations: Accuracy and stability problems

The initial value problem for Delay Differential Equations (DDEs)(1)y1(t)=f(t,y(t),y(t−τ1),…,y(t−τn)), t≥t0,y(t)=φ(t), t≤t0.is more complicated than the classical Cauchy problem from both theoretical and numerical point of view. In particular, the numerical integration of (1) requires the use of a continuous approximation such as, for instance, continuous Runge-Kutta methods. The need of continuous methods, rather than discrete methods, leads to new and difficult issues for the accuracy analysis and for the preservation of stability properties as well.

The asymptotic stability of multistep multiderivative methods for systems of delay differential equations

This paper deals with the asymptotic stability of multistep multiderivative methods for systems of delay differential equations. In particular, it is shown that A-stability of multistep multiderivative methods for ordinary differential equations is equivalent to P-stability of the induced methods for delay differential equations.

GP-Stability of two-step implicit runge-kutta methods for delay differential equations

The GP-stability of two-step implicit Runge-Kutta (TIRK) methods for the numerical solution of systems of delay differential equations (DDEs) is considered. We focus on the stability behaviour of TIRK methods in the solution of the linear constant-coefficient systems of DDE with a single delay. We present a sufficient condition of GP-stability of TIRK methods.

The p-stability and q-stability of singly diagonally implicit runge-kutta method for delay differential equations

The stability properties for singly diagonally implicit runge-kutta method fourth order five stages embedded in fifth six stages combined with lagrange interpolsation when applied to the linear delay differential equation. are investigated. Their regions of stability for are determined and numerical results when some problems are solved using the method are presented.

Stability of 2h-step spline method for delay differential equations

In this paper we consider a linear test equation to study the stability analysis of 2h-step spline method for the solution of delay differential equations. We prove that, this method is P-stable for cubic spline.

On the attainable order of collocation methods for delay differential equations with proportional delay

To analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods for the delay differential equation (DDE) y′(t) = by(qt), 0 < q ≤ 1 with y(0) = 1, and the delay Volterra integral equation (DVIE) y(t) = 1 + $$\tfrac{b}{q}\int {_0^{qt} }$$ y(s) ds with proportional delay qt, 0 < q ≤ 1, our particular interest lies in the approximations (and their orders) at the first mesh point t = h for the collocation solution v(t) of the DDE and the iterated collocation solution u it(t) of the DVIE to the solution y(t). Recently, H. Brunner proposed the following open problem: “For m ≤ 3, do there exist collocation points c i = c i(q), i = 1, 2,..., m in [0,1] such that the rational approximant v(h)is the (m, m)-Padé approximant to y(h)? If these exist, then |v(h) − y(h)| = O(h 2m+1) but what is the collocation polynomial M m(t; q) = K Π i=1 m (t − c i) of v(th), t ∈ [0, 1]?” In this paper, we solve this question affirmatively, and give the related results between the collocation solution v(t) of the DDE and the iterated collocation solution u it(t) of the DVIE. We also answer to Brunner's second open question in the case that one collocation point is fixed at the right end point of the interval.

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.

Synchronization of coupled time-delay systems: Analytical estimations

The synchronization threshold of coupled time-delay chaotic systems is estimated by two different analytical approaches. One of them is based on the Krasovskii-Lyapunov theory that represents an extension of the second Lyapunov method for delay differential equations. Another approach uses a perturbation theory of large delay time. The analytical expression relating synchronization threshold to the maximal Lyapunov exponent of uncoupled driving and response subsystems is derived. The analytical results are compared with the numerical simulations for two coupled Mackey-Glass systems.

Delay dependent stability regions of -methods for delay differential equations

In this paper asymptotic stability properties of Θ-methods for delay differential equations (DDEs) are considered with respect to the test equation {y'(t) = ay(t)+by(t-τ), t >0, (0.1) {y(t) = g(t), -τ≤t≤0 where τ> 0. First we examine extensively the instance where a, b ∈ R and g(t) is a continuous real-valued function; then we investigate the more general case of a, b ∈ C and g(t) a continuous complex-valued function. The last decade has seen a relatively large number of papers devoted to the study of the stability of Θ-methods, using the test equation (0.1). In those papers, conditions that are stronger than necessary for the (asymptotic) stability of the zero solution are assumed; for instance, R[a]+|b| 0. In this paper we study, instead, the stability properties of Θ-methods for equation (0.1) with an arbitrary but fixed value of τ.

Stability of the Radau IA and Lobatto IIIC methods for delay differential equations

Recently, we have proved that the Radau IA and Lobatto IIIC methods are P-stable, i.e., they have an analogous stability property to A-stability with respect to scalar delay differential equations (DDEs). In this paper, we study stability of those methods applied to multidimensional DDEs. We show that they have a similar property to P-stability with respect to multidimensional equations which satisfy certain conditions for asymptotic stability of the zero solutions. The conditions are closely related to stability criteria for DDEs considered in systems theory.

Variational iteration method for delay differential equations

The variational iteration method is shown to be applicable to delay differential equations for analytical approximate solutions.

Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations

This paper deals with the stability and convergence of Runge-Kutta methods with the Lagrangian interpolation (RKLMs) for nonlinear delay differential equations (DDEs). Some new concepts, such as strong algebraic stability, GDN-stability and D-convergence, are introduced. We show that strong algebraic stability of a RKM for ODEs implies GDN-stability of the corresponding RKLM for DDEs, and that a strongly algebraically stable and diagonally stable RKM with order p, together with a Lagrangian interpolation of order q, leads a D-convergent RKLM of order min{ p, q + 1}.

On the asymptotic stability properties of Runge-Kutta methods for delay differential equations

In this paper asymptotic stability properties of Runge-Kutta (R-K) methods for delay differential equations (DDEs) are considered with respect to the following test equation: \( \begin{eqnarray*} \left\{ \begin{array}{ccl} y'(t) & = & a\, y(t) + b\, y(t\,-\tau), \\[0.3cm] y(t) & = & g(t), \end{array} \right. \hskip 0.3cm \begin{array}{r} t > 0, \\[0.3cm] -\tau \leq t \leq 0, \end{array} \end{eqnarray*} \) where \(a,b \in {\Bbb R} ,\,\tau > 0\) and \(g(t)\) is a continuous real-valued function. In the last few years, stability properties of R-K methods applied to DDEs have been studied by numerous authors who have considered regions of asymptotic stability for “any positive delay” (and thus independent of the specific value of \(\tau\)).

Contractivity of continuous Runge-Kutta methods for delay differential equations

In this paper the author investigates the stability of numerical methods for general delay differential equations of the type ▪ where α( t) ≤ t and y( t) is a vector complex-valued function. Contractivity conditions are found for Runge-Kutta methods as applied to linear and nonlinear scalar equations. As for systems, a general condition is found for the contractivity of the solution of (1) in any vector norm, and a numerical method is proposed which preserves the contractivity in the maximum norm.

Stability analysis of continuous implicit Runge-Kutta methods for Volterra integro-differential systems with unbounded delays

In this paper we investigate conditions for the stability of solutions of variable-step continuous, implicit, Runge-Kutta-based methods for linear and nonlinear delay Volterra integro-differential equations (DVIDEs) with a “fading memory”. We present these methods as adaptations (based upon techniques in the treatment of Volterra integral equations) of corresponding collocation Runge-Kutta (CORK) methods for delay differential equations. The basic tool of our stability analysis, and a novel feature of the paper, is the introduction of and use of suitably-constructed discrete Lyapunov functionals, combined with a discrete difference inequality of unbounded order.

Regularity properties of Runge-Kutta methods for delay differential equations

Conditions are investigated which guarantee that Runge-Kutta methods preserve the asymptotic values of systems of delay differential equations. Such methods are said to be strongly regular. A constructive test for strong regularity is derived and examples of such methods are presented for s = p = 2 and s = p = 3, where s is the number of stages and p is the order of the method.

The stability of natural Runge-Kutta methods for nonlinear delay differential equations

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any collocation method is equivalent to one of such methods. In this paper, stability properties of natural Runge-Kutta methods are studied using nonlinear DDEs which have a quadratic Liapunov functional. A discrete analogue of the functional is defined for each method, and the stability of the method is examined on the basis of this analogue. In particular, it is shown that an algebracially stable method, if it satisfies an additional condition, preserves the asymptotic properties of the original equations for every stepsize.

The Numerical Stability of Linear Multistep Methods for Delay Differential Equations with Many Delays

This paper deals with the stability analysis of linear multistep methods for the numerical solution of delay differential equations (DDSs) with many delays. We focus on the stability behaviour of such methods when they are applied to a scalar, linear DDE, with many constant delays and complex coefficients. The concept of $GP_m $-stability is introduced. Under a sufficient condition for the test equation to be stable, the linear multistep method, when combined with a certain Lagrange interpolation, must be stable for all stepsizes. It is proven that a linear multistep method is $GP_m $-stable if and only if it is A-stable for ordinary differential equations (ODEs).

A stability property ofA-stable natural Runge-Kutta methods for systems of delay differential equations

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any one-step collocation method is equivalent to one of such methods. In this paper, we consider a linear constant-coefficient system of DDEs with a constant delay, and discuss the application of natural Runge-Kutta methods to the system. We show that anA-stable method preserves the asymptotic stability property of the analytical solutions of the system.

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

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