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**Collocation methods for Volterra integral and related functional differential equations.**
*(English)*
Zbl 1059.65122

Cambridge Monographs on Applied and Computational Mathematics 15. Cambridge: Cambridge University Press (ISBN 0-521-80615-1/hbk). xiv, 597 p. (2004).

Numerical methods for integral equations of Volterra type have been the subject of many investigations over the last decades. The book under review, written by one of the leading experts in this area, is on the one hand focused on a seemingly narrow part of this subject, namely methods of collocation type. On the other hand, it shows an enormous broadness because it covers not only the usual simple problems (such as equations with continuous or even smooth kernels) but also equations with weakly singular kernels, various forms of delay equations, integro-differential equations, integral-algebraic equations and equations with singular perturbations.

All these items are discussed in a thorough and very detailed fashion, including a review of the analytical aspects that are relevant for the numerical work, thus turning the monograph into a highly valuable resource for any researcher in the area. The style of the book is very similar to that of the older book written by the author and P. J. van der Houwen on a similar subject [The numerical solution of Volterra equations (North-Holland, Amsterdam) (1986; Zbl 0611.65092)]. Each section contains notes and exercises indicating possibilities for further research activities. The list of references is about 80 pages long and seems to be very complete.

All these items are discussed in a thorough and very detailed fashion, including a review of the analytical aspects that are relevant for the numerical work, thus turning the monograph into a highly valuable resource for any researcher in the area. The style of the book is very similar to that of the older book written by the author and P. J. van der Houwen on a similar subject [The numerical solution of Volterra equations (North-Holland, Amsterdam) (1986; Zbl 0611.65092)]. Each section contains notes and exercises indicating possibilities for further research activities. The list of references is about 80 pages long and seems to be very complete.

Reviewer: Kai Diethelm (Braunschweig)

### MSC:

65R20 | Numerical methods for integral equations |

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

45D05 | Volterra integral equations |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

45G10 | Other nonlinear integral equations |

45J05 | Integro-ordinary differential equations |