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

• Herbert Arndt
Chapter
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 62)

## Abstract

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].

## Literatur

1. 1.
Arndt, H. Zur Schrittweitensteuerung bei retardierten Anfangswertproblemen, to appear in ZAMM 63, Heft 4/5.Google Scholar
2. 2.
Driver, R.D. (1963) Existence Theory for a Delay-Differential System, Contributions to Differential Equations 1, 317–336.Google Scholar
3. 3.
Hale, J. (1977) Theory of Functional Differential Equations, Springer Verlag.Google Scholar
4. 4.
van der Houwen, P.J./Sommeijer, B.P. Improved Absolute Stability of Predictor-Corrector Methods for Retarded Differential Equations, these proceedings.Google Scholar
5. 5.
Oberle, H.J./ Pesch, H.J. (1981) Numerical Treatment of Delay Differential Equations by Hermite Interpolation, Numer.Math. 37 235–255.
6. 6.
Oppelstrup, J. (1976) The RKFHB4 Method for Delay Differential Equations, Springer Verlag, Lecture Notes in Mathematics 631, 133–146.Google Scholar
7. 7.
Shampine, L.F./ Watts, H.A. (1976) Global Error Estimation for Ordinary Differential Equations, ACM Trans. Math. Software 2, 172–186.
8. 8.
Stetter, H.J. (1965) Numerische Lösung von Differentialgleichungen mit nacheilendem Argument, ZAMM 45, T79–80.Google Scholar
9. 9.
Stetter, H.J. (1979) Global Error Estimation in Adams PC-Codes, ACM Trans. Math. Software 5, 415–430.
10. 10.
Tavernini, L. (1973) Linear Multistep Methods for the Numerical Solution of Volterra Differential Equations, Applicable Analysis 3, 169–185.
11. 11.
Tavernini, L. (1978) The Approximate Solution of Volterra Differential Systems with State-Dependent Time Lags, SIAM J. Numer. Anal. 15, 1039–1052.
12. 12.
Verwer, J.G. Estimation of the Global Error in Runge-Kutta Methods, these proceedings.Google Scholar