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.

Issues in the numerical solution of evolutionary delay differential equations

One-step collocation for delay differential equations

The subject of this paper is a variable stepsize one-step method of collocation type for solving general nonlinear second kind Volterra integral equations. We extend the iterated collocation method corresponding to polynomial spline collocation to nonlinear Volterra integral equations of the second kind. The resulting superconvergence properties of either the collocation approximation or the iterated collocation approximation are used to obtain (local and global) error estimates which in turn form the basis of a variable stepsize code. The performance of this code is illustrated by means of numerous test problems.

/pdf/the-numerical-solution-of-nonlinear-volterra-integral-42jmn2vt0n.pdf

The numerical solution of nonlinear volterra integral equations of the second kind by collacation and iterated collocation methods

We have recently developed a generic approach to solving retarded and neutral delay differential equations (DDEs) The approach is based on the use of an explicit continuous Runge--Kutta formula and employs defect control The approach can be applied to problems with state-dependent delays and vanishing delays
In this paper we determine convergence properties for the numerical solution associated with methods that implement this approach We first analyze such properties for retarded DDEs and then the analysis is extended to the case of neutral DDEs (NDEs) Such an extension is particularly important since NDEs have received little attention in the literature The main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect Our analysis can also be applied directly to other numerical methods (which use defect error control) based on Runge--Kutta formulas for DDEs

Convergence Analysis of the Solution of Retarded and Neutral Delay Differential Equations by Continuous Numerical Methods

/pdf/natural-continuous-extensions-of-runge-kutta-formulas-33usxj9dme.pdf

Natural continuous extensions of Runge-Kutta formulas

Retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme Hence one can control the global error of the modified problem but not directly the actual global error of the original problem In this paper we give an estimate for the actual global error in terms of controllable quantities Further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems Along with the new definition we are led to developing a reliable basis for a step selection scheme

Numerical solution of retarded initial value problems: Local and global error and stepsize control

In this paper we consider rational interpolation for an Hermite Problem, i.e. prescribed values of functionf and its derivatives. The algorithm presented here computes a solutionp/q of the linearized equationsp?fq=0 in form of a generalized continued fraction. Numeratorp and denominatorq of the solution attain minimal degree compatible with the linearized problem. The main advantage of this algorithm lies in the fact that accidental zeros of denominator calculated during the algorithm cannot lead to an unexpected stop of the algorithm. Unattainable points are characterized.

Ein verallgemeinerter Kettenbruch-Algorithmus zur rationalen Hermite-Interpolation

Dieses Buch uber gewohnliche Differentialgleichungen ist aus einem Kurs fur die Fernhochschule Hagen hervorgegangen, an dem seinerzeit auch Herr Dr. P. Janssen mitgearbeitet hatte. Ziel des damaligen Kurses war es, den Studenten, die ihrer Natur nach nicht so haufig personlichen Kontakt mit der Hochschule haben, eine Einfuhrung in die Theorie und Praxis der gewohnlichen Differentialgleichungen in einer Form zu geben, die dem hauslichen Selbststudium entgegenkommt. Dieses Konzept wurde auch in der vorliegenden Darstellung beibehalten. Jedoch kann man bei einer solchen Intention nicht erwarten, dass alle Aspekte der gewohnlichen Differentialgleichungen auch nur annahernd angesprochen werden konnen. Dem Leser, der an weiterfuhrenden Themen wie z.B. "Periodische Losungen" oder "Verzweigungstheorie" interessiert ist, soll das Literaturverzeichnis helfen. Leider hat Herr Werner die endgultige Fertigstellung dieses Buches nicht mehr erleben konnen - er verstarb im November 1985. Aus der engen Zusammenarbeit mit ihm kann ich jedoch sagen, dass ihm die Anfertigung dieses Buches mit seinem Thema und seiner didaktischen Ausrichtung ein besonderes Anliegen war und er bis zuletzt stets mit Freude daran gearbeitet h

Gewöhnliche Differentialgleichungen : eine Einführung in Theorie und Praxis

In dieser Arbeit werden nichtlineare Splines zur Losung von Anfangswertaufgaben bei gewohnlichen Differentialgleichungen herangezogen. In der Nahe von Singularitaten besitzen z.B. verallgemeinerte rationale Splines mit variablen Exponenten gute Approximationseigenschaften. Bei Polynomsplines konnen Konvergenzaussagen hergeleitet werden, indem Aquivalenz dieser Verfahren mit gewissen linearen Mehrschrittverfahren gezeigt wird. In dieser Arbeit behandeln wir den nichtlinearen Fall, indem wir die lokalen Fehler in den Knoten direkt verfolgen. Einige numerische Beispiele zeigen die Gute dieser Verfahren insbesondere bei solchen Losungen, die sehr steil anwachsen oder sogar im betrachteten Intervall singular werden.

We consider the technique of using nonlinear splines to solve the initial value problem of ordinary differential equations. It is known, for example, that generalized rational splines with variable exponents yield good approximations to the exact solution in the neighborhood of a singularity. In the case of polynomial splines, convergence results may be derived by demonstrating the equivalence of the method to linear multistep methods. This sort of analysis has been done by many authors. In this paper we treat the nonlinear case and are able to prove convergence by directly estimating the local errors at interior knots. Some computational examples are given which illustrate the power of the method near a singularity.

Lösung von gewöhnlichen Differentialgleichungen mit nichtlinearen Splines

Consider the retarded initial value problem 

$$\begin{gathered} y'(x) = f(x,y(x),y(x - \tau ))\quad for\quad x\underline > {x_o}, \hfill \\ y(x) = \psi (x)\quad \quad \quad \quad \quad \quad for\quad x\underline > {x_o}, \hfill \\ \end{gathered} $$

(1)

where the retardation τ may be constant, variable: τ = τ(x), or state dependent: τ = τ(x,y(x)), and τ ≥ o. If f, ψ and τ are continuous then according to DRIVER [2] there exists a solution of problem (1) in some interval [xo,b].

The Influence of Interpolation on the Global Error in Retarded Differential Equations

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