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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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Runge-Kutta Methods For DDEs

Runge-Kutta Methods For DDEs

(p.152) 6 Runge-Kutta Methods For DDEs
Numerical Methods for Delay Differential Equations

Alfredo Bellen

Marino Zennaro

Oxford University Press

This chapter specializes the standard approach presented in Chapter 4, based on the class of continuous Runge-Kutta methods reviewed in Chapter 5. The various cases of DDEs and neutral DDEs along the lines of Chapter 4 are reviewed, and further attention is given to the constant delay case for which the connection with Bellman's method of steps is illustrated. The non-vanishing, time-dependent delay case is investigated in detail, for which some additional results on superconvergence of constrained mesh methods are given, using the results on natural continuous extensions reported in Chapter 5. The numerical solution of the prototype of DDEs with vanishing delay, that is, the pantograph equation, is also investigated. RADAR5, the most recent code designed for a very general class of stiff delay differential-algebraic equations, is briefly introduced.

Keywords:   standard approach, Bellman's method, superconvergence, constrained mesh, constant delay, vanishing delay, pantograph equation

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