international journal of numerical modelling: electronic networks, devices and fieldsInt. J. Numer. Model.11, 231 (1998)

BOOK REVIEW

Modelling with Differential and DifferenceEquations by Glenn Fulford, Peter Forrester andArthur Jones, Australian Mathematical Society Lec-ture Series 10, Cambridge University Press, 1997.ISBN 0-521-44069-6 hardback, ISBN 0-521-44618-X paperback. No. of pages: 405. Price: £50 hard-back, £19.95 paperback.

This is what the cover says: ‘The real world canbe modelled using mathematics, and the constructionof such models is the theme of this book. Theauthors concentrate on the techniques used to setup mathematical models and describe many systemsin full detail, covering both differential and integralequations in depth. Among the broad spectrum oftopics studied are: mechanics, genetics, thermalphysics, economics and population studies. Anystudent wishing to solve problems via mathematicalmodelling will find that this text provides an excel-lent introduction to the subject.”

The book certainly fulfils these promises admir-ably, and the authors are to be warmly congratulated.The aspects of the physical world to be modelledare explained simply and clearly. In particular, thetreatment of mechanics is of such depth, clarity andextension that it could almost constitute a full text-book in its own right. Furthermore, the other topicsmentioned above also receive excellent treatment,being presented attractively and in sufficient depthto allow the student to proceed with confidence tothe modelling stage.

One of the great strengths of the book is theexcellent presentation of the two-way modellingtransition, from the material world to maths, andback. Again and again the authors guide the studentthrough the cyclic steps, from the real world to asimplified or idealized version, to the correspondingmathematical model, to ‘solving’ the model to getresults, and finally back to assessing and interpretingthese results in comparison with reality, before, ifnecessary, modifying the model or refining theassumptions.

Numerical examples are well integrated into thetext. The authors’ experience as teachers can beseen not least in the balance between practicalexamples and general principles. They are carefulnot to generalize too soon.

Are there any criticisms? Well, a few commentsrather than criticisms could be made.

Firstly, the book’s strength is its weakness. Thegeneral idea of mathematical modelling of physicalprocesses embraces most subjects in science andengineering undergraduate programmes. Yet a typi-cal undergraduate syllabus, at least of the kind this

1998 John Wiley & Sons, Ltd.

reviewer is familiar with (in engineering faculties),would require considerable reorganization beforesuch a book could constitute a core text. The bookstraddles (or integrates) significant parts of manytraditionally separate disciplines: mechanics, calcu-lus, numerical methods and sundry other sciences.While this may well be a virtue, it will neverthelesslimit the usefulness of this book to courses andinstitutions (arguably enlightened institutions) whereundergraduate courses in mathematical modelling arepresented as stand-alone courses. Lecturers in otherdisciplines will no doubt prefer other textbooksexclusively devoted to their particular area, wherethe modelling process is implied rather than cel-ebrated.

Secondly, on the topic of difference equations,the book gives many lovely examples of situationswhere difference equations give the best models.The peculiar behaviour of difference equations andtheir solution are also well treated, and some linksare made with differential equations in this context.

Initally, however, it seemed strange that probablythe most common use of difference equations is nothighlighted, namely, as discretized approximationsto corresponding differential equations for the benefitof the ubiquitous digital computer. Similarly, digitalsignal processing and its importance are not men-tioned. On the other hand, a textbook should notattempt to cover every topic, and it is refreshing tofind difference equations presented as fundamentallyimportant modelling devices in their own right ratherthan as (merely) approximate models of differentialequations (which, in turn, of course are ‘merely’approximate models of the physical world, a pointfrequently overlooked by numerical analysts, whooften present analytical solutions as ‘exact’ answers).

Other comments are minor. The use of the term‘antidifferentiation’, without even an explanatorycomment or footnote, seemed idiosyncratic to thisreviewer. Later in the book the word ‘integration’is used routinely, so the advantages of two namesfor the same thing are not obvious.

Finally, in going through the book a number oftypographical errors and at least one omitted refer-ence were discovered – a pity, although hard toavoid in a first edition.

In summary, the book is a delight and achievesits stated aims superbly, but its usefulness as anundergraduate text will depend largely on the teach-ing philosophy of the syllabus organisers.

William O’ConnorMechanical Engineering

University College DublinIreland