Collocation Methods for Volterra Integral and Related Functional Differential Equations

by Hermann Brunner

Overview

This is the first comprehensive introduction to collocation methods for the numerical solution of initial-value problems for ordinary differential equations, Volterra integral and integro-differential equations, and various classes of more general functional equations. It guides the reader from the "basics" to the current state-of-the-art level of the field, describes important problems and directions for future research, and highlights methods. The analysis includes numerous exercises and applications to the modelling of physical and biological phenomena.

Product Details

ISBN-13:	9780521806152
Publisher:	Cambridge University Press
Publication date:	11/15/2004
Series:	Cambridge Monographs on Applied and Computational Mathematics , #15
Pages:	612
Product dimensions:	6.22(w) x 9.29(h) x 1.57(d)

Table of Contents

1. The collocation method for ODEs: an introduction; 2. Volterra integral equations with smooth kernels; 3. Volterra integro-differential equations with smooth kernels; 4. Initial-value problems with non-vanishing delays; 5. Initial-value problems with proportional (vanishing) delays; 6. Volterra integral equations with weakly singular kernels; 7. VIDEs with weakly singular kernels; 8. Outlook: integral-algebraic equations and beyond; 9. Epilogue.

Editorial Reviews

"[T]his book gives a rather comprehensive treatment of collocation methods and its application to a wide class of functional equations. Even though it is centred on the use of collocation, this book also provides an introductory survey on theoretical and practical problems related to several kinds of Volterra Functional Equations and their numerical integration. The clarity of the exposition, the completeness in the presentation of stated and proved theorems, and the inclusion of a long list of exercises and open problems, along with a wide and exhaustive annotated bibliography, make this monograph a useful and valuable reference book for a wide range of scientists and engineers. In particular, it can be recommended to advanced undergraduate and graduate students in mathematics and may also serve as a source of topics for M.Sc. and Ph.D. theses in this field."
Journal of Integral Equations and Applications

"The book gives a state-of-the-art view of the numerical solution of Volterra equations and opens a rich source of unsolved problems for future research."
Mathematical Reviews, G.A. Evans

"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."
Zentralblatt MATH

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