Methods For Delay Differential Equations Research Articles

Projective synchronization of time‒delayed chaotic systems with unknown parameters using adaptive control method

The present article aims to study the projective synchronization between two identical and non‒identical time‒delayed chaotic systems with fully unknown parameters. Here the asymptotical and global synchronization are achieved by means of adaptive control approach based on Lyapunov–Krasovskii functional theory. The proposed technique is successfully applied to investigate the projective synchronization for the pairs of time‒delayed chaotic systems amongst advanced Lorenz system as drive system with multiple delay Rössler system and time‒delayed Chua's oscillator as response system. An adaptive controller and parameter update laws for unknown parameters are designed so that the drive system is controlled to be the response system. Numerical simulation results, depicted graphically, are carried out using Runge–Kutta Method for delay‒differential equations, showing that the design of controller and the adaptive parameter laws are very effective and reliable and can be applied for synchronization of time‒delayed chaotic systems. Copyright © 2014 John Wiley & Sons, Ltd.

Stepsize Restrictions for Nonlinear Stability Properties of Neutral Delay Differential Equations

The present paper is concerned with the relationship between stepsize restriction and nonlinear stability of Runge-Kutta methods for delay differential equations. We obtain a special stepsize condition guaranteeing global and asymptotical stability properties of numerical methods. Some confirmations of the conditions on Runge-Kutta methods are illustrated at last.

The $hp$ Discontinuous Galerkin Method for Delay Differential Equations with Nonlinear Vanishing Delay

We present the $hp$-version of the discontinuous Galerkin method for the numerical solution of delay differential equations with nonlinear vanishing delays and derive error bounds that are explicit in the time steps, the degrees of the approximating polynomials, and the regularity properties of the exact solutions. It is shown that the $hp$ discontinuous Galerkin method exhibits exponential rates of convergence for smooth solutions on uniform meshes, and for nonsmooth solutions on geometrically graded meshes. The theoretical results are illustrated by various numerical examples.

The long time behavior of a spectral collocation method for delay differential equations of pantograph type

In this paper, we propose an efficient numerical method for delay differential equations with vanishing proportional delay qt (0 < q < 1). The algorithm is a mixture of the Legendre-Gauss collocation method and domain decomposition. It has global convergence and spectral accuracy provided that the data in the given pantograph delay differential equation are sufficiently smooth. Numerical results demonstrate the spectral accuracy of this approach and coincide well with theoretical analysis.

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

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.

Stability and convergence of the two parameter cubic spline collocation method for delay differential equations

In this paper, we propose the cubic spline collocation method with two parameters for solving delay differential equations (DDEs). Some results of the local truncation error and the convergence of the spline collocation method are given. We also obtain some results of the linear stability and the nonlinear stability of the method for DDEs. In particular, we design an algorithm to obtain the ranges of the two parameters α , β which are necessary for the P-stability of the collocation method. Some illustrative examples successfully verify our theoretical results.

Formula omitted]-stability of the split-step [formula omitted]-methods for linear stochastic delay integro-differential equations

In this paper, T (Trajectory)-stability of the split-step θ -methods for linear stochastic delay integro-differential equations is studied. The split-step θ -methods for stochastic differential equations were introduced in Ding et al. (2010) [18] and the T -stability of the semi-implicit Euler method for delay differential equations with multiplicative noise has recently been discussed in Cao (2010) [17]. Motivated by the work of Ding et al. (2010) [18] and Cao (2010) [17], we investigate the T -stability of the split-step θ -methods for linear stochastic delay integro-differential equations. The Wiener increment is approximated by a discrete random variable with two-point distribution. Numerical experiments are also provided to illustrate the theory.

Stability analysis of exponential Runge–Kutta methods for delay differential equations

A sufficient condition of stability of exponential Runge–Kutta methods for delay differential equations is obtained. Furthermore, a relationship between P-stability and GP-stability is established. It is proved that the numerical methods can preserve the analytical stability for a class of test problems.

A Parallel Iteration of Runge-Kutta Method for Delay Differential Equations

Article A Parallel Iteration of Runge-Kutta Method for Delay Differential Equations was published on December 1, 2010 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 11, issue Supplement).

Variational Iteration Method for Delay Differential Equations Using He’s Polynomials

January 21, 2010 In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving delay differential equations which are otherwise too difficult to solve. These equations arise very frequently in signal processing, digital images, physics, and applied sciences. Numerical results reveal the complete reliability and efficiency of the proposed combination.

Stability of collocation methods for delay differential equations with vanishing delays

We analyze the asymptotic stability of collocation solutions in spaces of globally continuous piecewise polynomials on uniform meshes for linear delay differential equations with vanishing proportional delay qt (0<q<1) (pantograph DDEs). It is shown that if the collocation points are such that the analogous collocation solution for ODEs is A-stable, then this asymptotic behaviour is inherited by the collocation solution for the pantograph DDE.

Exponential Runge–Kutta methods for delay differential equations

This paper deals with convergence and stability of exponential Runge–Kutta methods of collocation type for delay differential equations. It is proved that these kinds of numerical methods converge at least with their stage order. Moreover, a sufficient condition of the numerical stability is provided. Finally, some numerical examples are presented to illustrate the main conclusions.

Block boundary value methods for delay differential equations

In this paper, a class of block boundary value methods (BBVMs) for the initial value problems of delay differential equations are suggested. It is proven under the classical Lipschitz condition that a BBVM is convergent of order p if it is consistent of order p. Several linear stability criteria for the BBVMs are derived. Numerical experiments further confirm the convergence and the effectiveness of the methods.

T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise

The paper deals with the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. A difference equation is obtained by applying the numerical method to a linear test equation, in which the Wiener increment is approximated by a discrete random variable with two-point distribution. The conditions under which the method is T-stable are considered and the numerical experiments are given.

Discontinuous Galerkin Methods for Delay Differential Equations of Pantograph Type

This paper is concerned with the application of the discontinuous Galerkin method to delay differential equations with vanishing delay $qt$ ($0<q<1$). Our aim is to establish optimal global and local superconvergence results on uniform meshes and compare these with analogous estimates for collocation methods. The theoretical results are illustrated by a broad range of numerical examples.

Nonlinear stability of discontinuous Galerkin methods for delay differential equations

The present paper is devoted to a study of nonlinear stability of discontinuous Galerkin methods for delay differential equations. Some concepts, such as global and analogously asymptotical stability are introduced. We derive that discontinuous Galerkin methods lead to global and analogously asymptotical stability for delay differential equations. And these nonlinear stability properties reveal to the reader the relation between the perturbations of the numerical solution and that of the initial value or the systems.

Delay-dependent stability analysis of multistep methods for delay differential equations

This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, linear multistep methods are considered and a class of τ(0)-stable methods are found. Later, some examples of τ(0)-stable multistep multistage methods are given. Finally, numerical experiments are presented to confirm the theoretical results.

Stability of implicit-explicit linear multistep methods for ordinary and delay differential equations

Stability properties of implicit-explicit (IMEX) linear multistep methods for ordinary and delay differential equations are analyzed on the basis of stability regions defined by using scalar test equations. The analysis is closely related to the stability analysis of the standard linear multistep methods for delay differential equations. A new second-order IMEX method which has approximately the same stability region as that of the IMEX Euler method, the simplest IMEX method of order 1, is proposed. Some numerical results are also presented which show superiority of the new method.

Delay-dependent stability of high order Runge–Kutta methods

This paper is concerned with the study of the delay-dependent stability of Runge–Kutta methods for delay differential equations. First, a new sufficient and necessary condition is given for the asy...

Synchronization threshold of a coupled n-dimensional time-delay system

The synchronization threshold in the general form of one way time-delay system is discussed. The synchronization threshold of coupled time-delay chaotic systems can be 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. Based on the Krasovskii–Lyapunov theory, the deduction process and the application range of the synchronization threshold are given.