Numerical Methods for Delay Differential Equations

Hardcover | December 9, 2003

byAlfredo Bellen, Marino Zennaro

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The main purpose of the book is to introduce the readers to the numerical integration of the Cauchy problem for delay differential equations (DDEs). Peculiarities and differences that DDEs exhibit with respect to ordinary differential equations are preliminarily outlined by numerous examplesillustrating some unexpected, and often surprising, behaviours of the analytical and numerical solutions. The effect of various kinds of delays on the regularity of the solution is described and some essential existence and uniqueness results are reported. The book is centered on the use ofRunge-Kutta methods continuously extended by polynomial interpolation, includes a brief review of the various approaches existing in the literature, and develops an exhaustive error and well-posedness analysis for the general classes of one-step and multistep methods. The book presents a comprehensive development of continuous extensions of Runge-Kutta methods which are of interest also in the numerical treatment of more general problems such as dense output, discontinuous equations, etc. Some deeper insight into convergence and superconvergence of continuousRunge-Kutta methods is carried out for DDEs with various kinds of delays. The stepsize control mechanism is also developed on a firm mathematical basis relying on the discrete and continuous local error estimates. Classical results and a unconventional analysis of "stability with respect to forcingterm" is reviewed for ordinary differential equations in view of the subsequent numerical stability analysis. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding stability requirements for the numerical methods are assessed andinvestigated. Alternative approaches, based on suitable formulation of DDEs as partial differential equations and subsequent semidiscretization are briefly described and compared with the classical approach. A list of available codes is provided, and illustrative examples, pseudo-codes and numerical experimentsare included throughout the book.Series Editors: G. H. Golub (Stanford University)C. Schwab (ETH Zurich)W. A. Light (University of Leicester)E. Suli (University of Oxford)Recent developments in the field of numerical analysis have radically changed the nature of the subject. Firstly, the increasing power and availability of computer workstations has allowed the widespread feasibility of complex numerical computations, and the demands of mathematical modelling areexpanding at a corresponding rate. In addition to this, the mathematical theory of numerical mathematics itself is growing in sophistication, and numerical analysis now generates research into relatively abstract mathematics.Oxford University Press has had an established series Monographs in Numerical Analysis, including Wilkinson's celebrated treatise The Algebraic Eigenvalue Problem. In the face of the developments in the field this has been relaunched as the Numerical Mathematics and Scientific Computation series. Asits name suggests, the series will now aim to cover the broad subject area concerned with theoretical and computational aspects of modern numerical mathematics.

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From the Publisher

The main purpose of the book is to introduce the readers to the numerical integration of the Cauchy problem for delay differential equations (DDEs). Peculiarities and differences that DDEs exhibit with respect to ordinary differential equations are preliminarily outlined by numerous examplesillustrating some unexpected, and often surpr...

Professor ALFREDO BELLEN Dipartimento di Scienze Matematiche. Universita' di Trieste, via Valerio 12/1, 34100 Trieste, Italy ++39 040 558 2608 ++39 040 558 2636 bellen@univ.trieste.it Full Professor of Numerical Analysis in Faculty of Engineerings. Department of Mathematical Sciences. University of Trieste, Italy Italian. Bo...

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Format:HardcoverDimensions:410 pages, 9.21 × 6.14 × 1.01 inPublished:December 9, 2003Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198506546

ISBN - 13:9780198506546

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Table of Contents

1. Introduction2. Existence and regularity of solutions of DDEs3. A review of DDE methods4. The standard approach via continuous ODE methods5. Continuous Runge-Kutta methods for ODEs6. Runge-Kutta methods for DDEs7. Local error estimation and variable stepsize8. Stability analysis of Runge-Kutta methods for ODEs9. Stability analysis of DDEs10. Stability analysis of Runge-Kutta methods for DDEs

Editorial Reviews

..."this is a serious monograph...It will serve as a guide and a reference for researchers and engineers interested in numerical solution of models from real-life applications. It may also serve as a textbook for a graduate-level course in this area that may be offered at some universities."--SIAM
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