UK Nonlinear News , November 1998

Mathematical Models in the Applied Sciences

By A.C. Fowler

Reviewed by Dr. Rob Seymour (Department of Mathematics, UCL)

Published by Cambridge University Press, 1997,
Paperback: ISBN 0 521 46703 9, Price: #24.95,
Hardback: 0 521 46140 5, Price: #65.00

The skill and art of mathematical modelling is notoriously difficult to teach, depending as it does on a mysterious mixture of technical knowledge, physical insight and ‘knowingness’. It is this last quality for which it is difficult to devise systematic and formal methods of instruction. There is no substitute for hard-won experience.

Perhaps the root of the problem is that ‘mathematical modelling’ is not really a subject in its own right, in spite of the British tradition of regarding ‘applied mathematics’ as essentially synonymous with fluid mechanics. First and foremost a modeller needs to be, or to collaborate with, and expert in some scientific discipline. Only with a developed understanding of appropriate scientific principles and how they apply to real-world situations, is it possible to write down a good mathematical model of some circumstance of interest. Such a model should represent all (apparently) relevant processes, but in as economical and unfussy a way as possible, yet without doing too much violence to what is known about how such processes operate. It is this judicious blend of precision with appropriate simplification which is hard to get right, hard to learn, and even harder to teach.

However, the applied mathematician also has to know what to do with a model when it has been constructed. Serious real-world models can present formidable analytical and computational challenges, which themselves more often than not require a high level of technique and experience to meet. Knowing what kind of information can be extracted from what kind of model, and how, is an important ingredient in the ‘knowingness’ which ideally should also be fed into the construction of the model in the first place. One should always ask: What do I want this model to do for me; what questions do I want to ask of it, and how do I design it with a reasonable prospect of getting answers?

There is a dual aspect to mathematical modelling, namely construction and analysis, which can, to a large extent, be separated. Thus, a textbook caricature might feature a scientist trained in the relevant principles of her subject, who formulates a model, while a mathematician, trained in the techniques of analysis, extracts the payoff from the model, with the scientist watching over his shoulder ready to offer interpretation of the analytical output. In practice, the process is messier than this cartoon picture, with plenty of overlap between the two roles.

So, what to do to learn mathematical modelling? There are really only three responses. First is to become a scientist of some recognisable variety, hopefully with some mathematical training thrown in (biologists please take note), and learn how scientific insight can be used to construct mathematical models within the chosen discipline. Second, become an applied mathematician, and hone your skills in the tricks of the analytical trade, without being very concerned about the genesis of the models you analyse. Third, become a jack of all trades (with the risk of being master of none), and ply your trade in both model building and analysis. For this ‘third way’, a thick skin is a desirable extra to protect against the risk not being taken seriously as either a scientist or a mathematician.

Authors of books on mathematical modelling must decide which of these potential audiences they wish to address. Books for scientists should concern themselves with the principles of a particular discipline, and books for mathematicians should focus primarily on the intricacies of analytical and/or computational methods. The last, jacks-of-all-trades audience, is probably the hardest to write for, but it is the audience which Andrew Fowler seems to have in mind in this book.

The technique chosen by the author is to present the reader with a wide range of models of varying levels of complexity from a number of different scientific disciplines. However, all the models have in common that they are deterministic with continuous space and time variables. That is, they are represented by systems of (non-linear) ordinary or partial differential equations. You will not find any discrete-time or stochastic models here. Many subject areas are visited along the way, including models of heat transfer, viscous flow, solid mechanics, electromagnetism, chemical reactions, population dynamics, river and ground water flow, convection in porous media and frost heave in freezing soils. A rather narrow repertoire of standard analytical techniques is applied to illustrate the uniformity of approach which can be brought to bear on models of this general type. The author rightly emphasises the importance of working with dimensionless variables - a process he calls non-dimensionalisation - which facilitates the identification of (relatively) fast and slow changing variables, and large and small length scales. This is the basis on which complex models can be partially decomposed and simplified, and is very much an art in its own right. Then the heavy artillery of asymptotic expansions and singular perturbation theory can be brought to bear to construct approximate solutions and extract relevant insights. The worked examples in the text are supplemented by substantial exercises at the end of each chapter.

This book is not for novices, in spite of its claim on the back cover that only a basic mathematical grounding in calculus and analysis is assumed. Indeed, the book is based on a graduate course given by the author at the University of Oxford. The explication of both the analytical methods used and the various physical principles from which models are derived, is cursory to say the least. This is action-man’s book, impatient with formal theory and keen to engage with the ‘real world’. The didactic strategy is to illustrate analysis-in-action, and to hope that the all-important ‘knowingness’ will self assemble out of the student’s struggle to comprehend the sketches given. So, rich as it is in examples, experience and insight, I would recommend that the student interact with a good, warm-blooded teacher to help with the reading-between-the-lines which is very necessary to get the most out of this book. The effort could be richly rewarding, in spite of the minor irritation a British reader may feel at the American spelling of ‘modelling’ throughout.

UK Nonlinear News thanks Cambridge University Press for providing a copy of this book for review.

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