Numerical methods for delay differential equations.

*(English)*Zbl 1038.65058
Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press (ISBN 0-19-850654-6/hbk). xiv, 395 p. (2003).

This is the first monograph devoted entirely to the numerical solution of delay differential equations (DDEs). Written by the researchers who have made essential contributions to this area over the past several years, the book attempts to provide up to date account of the theory and practice of the numerical solution of DDEs, where the right-hand side of the equation may depend on the history of the solution, and of neutral delay differential equations (NDDEs) where the right-hand side of the equation may also depend on the history of the derivative of the solution.

The presence of the initial function and the delay terms, which itself may depend on the unknown solution (the so-called state dependent delays), leads to a wide range of interesting and often surprising phenomena which are not encountered in the theory of ordinary differential equations (ODEs). Also depending on the behaviour of the lag terms the design of the appropriate methods for the numerical solution of delay differential equations and neutral delay differential equations can lead to new challenges which are not encountered in the numerical solution of ODEs. These challenges and the differences between the approaches to the numerical solution of DDEs and NDDEs and ODEs are discussed by means of many illustrative examples and experiments whose aim is to convince the reader that ‘integration of DDEs cannot be based on the mere adaptation of some standard ODE code to the presence of delayed terms. Integration of DDEs actually requires the use of specifically designed methods, according to the nature of the equation and the behaviour of the solution’.

Many convergence results are carefully formulated and rigorously proved under different assumptions on the right-hand side of the equation and on the delay terms. The authors also discuss the estimation of local discretization errors which form a basis for adaptive implementation of numerical methods on nonuniform meshes. The authors also explore the wealth of phenomena related to the stability analysis of numerical methods for ordinary and delay differential equations.

This book will serve as a guide and a reference for researchers and engineers interested in numerical solution of models from real-life applications based on these equations. It may also serve as a texbook for a graduate level course in this area which may be offered at some Universities.

The presence of the initial function and the delay terms, which itself may depend on the unknown solution (the so-called state dependent delays), leads to a wide range of interesting and often surprising phenomena which are not encountered in the theory of ordinary differential equations (ODEs). Also depending on the behaviour of the lag terms the design of the appropriate methods for the numerical solution of delay differential equations and neutral delay differential equations can lead to new challenges which are not encountered in the numerical solution of ODEs. These challenges and the differences between the approaches to the numerical solution of DDEs and NDDEs and ODEs are discussed by means of many illustrative examples and experiments whose aim is to convince the reader that ‘integration of DDEs cannot be based on the mere adaptation of some standard ODE code to the presence of delayed terms. Integration of DDEs actually requires the use of specifically designed methods, according to the nature of the equation and the behaviour of the solution’.

Many convergence results are carefully formulated and rigorously proved under different assumptions on the right-hand side of the equation and on the delay terms. The authors also discuss the estimation of local discretization errors which form a basis for adaptive implementation of numerical methods on nonuniform meshes. The authors also explore the wealth of phenomena related to the stability analysis of numerical methods for ordinary and delay differential equations.

This book will serve as a guide and a reference for researchers and engineers interested in numerical solution of models from real-life applications based on these equations. It may also serve as a texbook for a graduate level course in this area which may be offered at some Universities.

Reviewer: Z. Jackiewicz (Tempe)

##### MSC:

65L05 | Numerical methods for initial value problems |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |

34K40 | Neutral functional-differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65L70 | Error bounds for numerical methods for ordinary differential equations |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |