Numerical Methods for Delay Differential Equations (Numerical Mathematics and Scientific Computation)

The mainpurpose 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 examples illustrating some unexpected, and ofter surprising behaviors 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 various approaches existing in the literatire, and develps an exhaustive error and well-posedness analysis for the general classes of one-step and multistep methods. 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 examples illustrating 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 of Runge-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 continuous Runge-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 forcing term'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 and investigated. 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 experiments are included throughout the book.

for Mathematicians, Scientists, And Engineers Interested In The Theoretical Aspects Of Numerical Methods For Ordinary And Delay Differential Equations, Or Are Just Looking For A Suitable Technique To Simulate A Model By Numerically Solving Some Specific Equations, Bellen And Zennaro (both Mathematical Science, U. Of Trieste, Italy) Examine Initial Value Problems For Bolterra Functional Differential Equations Characterized By The Presence Of Discretely Distributed Delays, Possibly Of Neutral Type, Usually Called Delay Differential Equations. They Describe, Analyze, Unify, And When Necessary Extend And Improve The Various Approaches And Techniques That Appear In The Literature. Annotation ©2003 Book News, Inc., Portland, Or