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Numerical Methods for Delay Differential Equations$
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Alfredo Bellen and Marino Zennaro

Print publication date: 2003

Print ISBN-13: 9780198506546

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506546.001.0001

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The Standard Approach via Continuous ODE Methods

The Standard Approach via Continuous ODE Methods

Chapter:
(p.62) 4 The Standard Approach via Continuous ODE Methods
Source:
Numerical Methods for Delay Differential Equations
Author(s):

Alfredo Bellen

Marino Zennaro

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506546.003.0004

This chapter focuses on the error and convergence analysis of the technique based on the use of a discrete method for ODEs endowed with some continuous extension, which is called the standard approach. The technique is described for some of the main types of DDEs and neutral DDEs, ranging from constant or non-vanishing time dependent delays, to arbitrary time dependent delays, up to state dependent delays. It is shown that for time dependent and state dependent delays, the resulting method may become implicit even if the underlying ODE method is explicit and, even in this case, well-posedness is proven. The non-trivial problem of tracking the discontinuities is also considered. It is shown that despite the fact that any discrete method is, in principle, suitable for the standard approach, one-step methods (essentially Runge-Kutta methods) are preferable to multistep methods. A list of available codes is given.

Keywords:   error analysis, convergence analysis, standard approach, well-posedness, discontinuities, codes

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