Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs. Communicated by J.C. Mason     

Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs. Communicated by J.C. Mason     

An efficient unified approach for the numerical solution of delay differential equations

In this paper we propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a standard step-by-step approach, such as Runge-Kutta or linear multi-step methods, and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. Using this interpretation we develop an efficient technique to solve the resulting discontinuous IVP. We also give a more clear process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. The new modular design for the resulting simulator we introduce, helps to accelerate the utilization of advances in the different components of an effective numerical method. Such components include the underlying discrete formula, the interpolant for dense output, the strategy for handling discontinuities and the iteration scheme for solving any implicit equations that arise.     

Computing breaking points in implicit delay differential equations

Systems of implicit delay differential equations, including state-dependent problems, neutral and differential-algebraic equations, singularly perturbed problems, and small or vanishing delays are considered. The numerical integration of such problems is very sensitive to jump discontinuities in the solution or in its derivatives (so-called breaking points). In this article we discuss a new strategy – peculiar to implicit schemes – that allows codes to detect automatically and then to compute very accurately those breaking points which have to be inserted into the mesh to guarantee the required accuracy. In particular for state-dependent delays, where breaking points are not known in advance, this treatment leads to a significant improvement in accuracy. As a theoretical result we obtain a general convergence theorem which was missing in the literature (see Bellen and Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003). Furthermore, as a useful by-product, we design strategies that are able to detect points of non-uniqueness or non-existence of the solution so that the code can terminate when such a situation occurs. A new version of the code RADAR5 together with drivers for some real-life problems is available on the homepages of the authors. Supported by the Swiss National Science Foundation, project # 200020-101647.     

Hopf bifurcation in numerical approximation for delay differential equations

In this paper we investigate the qualitative behaviour of numerical approximation to a class delay differential equation. We consider the numerical solution of the delay differential equations undergoing a Hopf bifurcation. We prove the numerical approximation of delay differential equation had a Hopf bifurcation point if the true solution does.     

QMC Methods for the solution of delay differential equations

In this paper we will discuss the application of so‐called Runge Kutta Quasi‐Monte Carlo (RKQMC) methods (as proposed by Lécot, Koudiraty, Coulibaly, and Stengle) to heavily oscillating di.erential equations. The delayed argument is approximated by Hermite interpolation which transforms the equation into an ordinary differential equation so that the RKQMC methods can be applied. We will give a short discussion of this method and its advantages as well as its drawbacks, and give some more numerical results.     

Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method

Summary In this paper we develop a class of numerical methods to approximate the solutions of delay differential equations. They are essentially based on a modified version, in a predictor-corrector mode, of the one-step collocation method atn Gaussian points. These methods, applied to ODE's, provide a continuous approximate solution which is accurate of order 2n at the nodes and of ordern+1 uniformly in the whole interval. In order to extend the methods to delay differential equations, the uniform accuracy is raised to the order 2n by some a posteriori corrections. Numerical tests and comparisons with other methods are made on real-life problems.This work was supported by CNR within the Progetto Finalizzato Informatica-Sottopr. P1-SOFMAT     

The numerical solution of forward-backward differential equations: Decomposition and related issues

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x^′(t)=ax(t)+bx(t−1)+cx(t+1) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation of the solution. We include illustrative examples to demonstrate the success of our method, along with an indication of its current limitations.     

Methods for the solution ofAXD−BXC=E and its application in the numerical solution of implicit ordinary differential equations

The solution of the equationAXD–BXC=E is discussed, partly in terms of the generalized eigenproblem. Useful applications arise in connection with the numerical solution of implicit differential equations.     

Stability analysis of numerical methods for delay differential equations

Summary 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)+U(t–) with >0 and , complex. A general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods.     

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.     

Variational splines for the numerical solution of singular differential equations

The numerical parametrization method (PM), originally created for optimal control problems, is specificated for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs). The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous block implicit hybrid one-step methods for ordinary and delay differential equations

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and Aω-stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.     

线性时滞微分方程稳定性分析和数值例子

§ 1　IntroductionFunctional differential equations have a wide range of applications in science andengineering.The simplestand perhapsmostnatural type of functional differential equationis a“delay differential equation”,that is,differential equation with dependence on the paststate.The simplest type of pastdependence is thatit is carried through the state variablebut not through its derivative.Then the equation can be expressed as delay differentialequations(DDEs) .There are also a number…     

Asymptotic expansion for the solution of singularly perturbed delay differential equations

Singular perturbation problems occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. For practical problems, one seeks a uniformly valid, readily interpretable approximation to a solution that does not behave uniformly. In this paper we extend singular perturbation theory in ordinary differential equations to delay differential equations with a fixed lag. We aim to give an explicit sufficient condition so that the solution of a class of singularly perturbed delay differential equations can be asymptotically expanded. O'Malley-Hoppensteadt technique is adopted in the construction of approximate solutions for such problems. Some particular phenomena different from singularly perturbed ordinary differential equations are discovered.     

Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay

In this note we propose a method for the integration of y'(t) = f(t, y(t), y(rt)), 0 t t_f y(0) = y₀, where 0 < r < 1, by a superconvengent s-stage continuousRK method of discrete global order p and continuous uniformorder q < p – 1 for the approximation of the delayedterm y(rt). We prove that, although the maximum attainable orderof the method on an arbitrary mesh is q' = min{p, q + 1}, byusing a quasi-geometric mesh, introduced by Bellen et al. (1997,Appl. Numer. Math. 24, 1997, 279–293), the optimal accuracyorder p is preserved.