Mathematical Modeling

The goal of this book (an English translation of a German text first published about ten years ago) is to teach undergraduate and graduate students the basic examples and techniques of mathematical modeling of real-world phenomena. The phenomena discussed in the text are (mostly) continuous in nature and the mathematics used is primarily ordinary and differential equations.

There are seven chapters, arranged in increasing order of sophistication. Two of these seven chapters (specifically, the third and fifth) address areas of physics (thermodynamics and continuum mechanics) that will be shortly be discussed in the text. The first chapter is introductory; it starts by talking about what mathematical modeling is all about and then gives some extended “case studies” — population growth (where it is explained why the discrete model is best replaced by a continuous one) and an example from fluid mechanics. Techniques such as dimensional analysis and asymptotic expansions are also introduced and discussed.

This leaves chapters 2, 4, 6 and 7. Chapter 2 uses systems of linear equations to discuss equilibrium problems in a variety of different contexts: electrical networks, stress loads, etc. Chapter 4 discusses a number of models amenable to study by ordinary differential equations, including oscillations, pendulums, phase planes, and predator-prey models. Finally, chapters 6 and 7 concern partial differential equations. The first of these chapters discusses elliptic, hyperbolic and parabolic PDEs, though at a much higher level than is typically encountered in an undergraduate PDE text. The second and final chapter looks at “free boundary problems”; i.e., problems in which a PDE is defined on a domain, part of whose boundary is unknown. This chapter shows how these problems can be used to model melting problems, crystal growth, and contact problems involving elastic membranes.

This is a demanding book, one that expects a lot of the students who will be reading it. (This is likely related to its European origins.) The specific prerequisites tend to increase as the book progresses, but a basic background in real analysis and linear algebra is assumed throughout. Later chapters require some familiarity with ordinary and partial differential equations, the integral theorems of multivariable calculus, and the theory of surfaces. Some prior knowledge of physics would seem to be essential, as well.

Each chapter ends with exercises, many of which (to my inexpert eye, anyway) seemed to fall in the medium-to-difficult range. Most textbooks in the Springer Undergraduate Mathematics Series (SUMS) offer in-text solutions to at least some of the exercises (in fact, that’s one of the features advertised by Springer in their online description of the series) but this is one of the rare entries in that series that does not; there are no solutions to be found, either in the text itself or (as of this writing, anyway) on the webpage for the text.

The placement of this book in the SUMS series is, I think, unusual for other reasons as well. I have already noted that this is a demanding book. It certainly seems not only much larger, but also more sophisticated, than most, if not all, of the other SUMS books; in fact, I view this book as better addressed to graduate students than undergraduates. I don’t imagine, for example, that this book would be considered accessible by an average mathematics major at my university. However, for a very well-prepared reader with a willingness to work hard, there is a wealth of interesting material to be found here.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.