Mathematical Models and Their Analysis

This entry in SIAM’s Classics in Applied Mathematics series is a reprint of a book published in 1989. (It is essentially the same as the earlier edition but adds a short appendix on controllability of linear systems.) The book’s premise is that even very simple mathematical modeling and the analysis associated with it can reveal a good deal about real-life phenomena in engineering and the sciences well before computation and numerical simulations are brought into play.

The author’s preface to this new edition says that the previous publisher had suggested publishing a second edition twenty years ago provided the author included new material on “computer modeling”. He declined because he felt that this ran counter to the intent of his book and would dilute the message. When he reviewed the material again in preparation for this new edition he found it still fresh and relevant. Indeed, the book has aged pretty well and remains a valuable resource.

This book is organized around what the author calls “processes”; these are broad categories that include dynamical systems and their evolution, stability of equilibrium configurations, wave propagation, diffusion, control and optimization. The author suggests that these processes are generic and can be identified in a great many models with very disparate scientific origins. A separate section of the book is devoted to each process; each section then investigates two or more models of that type. Subjects for these models range from physics to mechanical engineering, traffic flow, economics and more. The last five chapters of the book address aspects of natural resource modeling that include fishing, timber and social equity questions.

One of the real strengths of the book is the depth of experience teaching mathematical modeling that the author displays. He understands the difficulties — that recipes for modeling don’t exist, that real life situations are complex and don’t lend themselves to quick success, and that the science required for good modeling varies a lot from problem to problem. He really understands how and why students find it difficult.

All the same, the author maintains that his book is not designed to teach mathematical modeling, but to identify and illustrate different kinds of mathematical models. He demonstrates an approach with a formulation-analysis-evaluation-interpretation-reformulation cycle in his case studies. Probably the best representative of that here is a succession of traffic models with repeated improvements in Chapters 6, 7 and 10.

The treatment of traffic models is typical of the author’s approach. He begins with a discussion of Eulerian (continuum) and Lagrangian (particle) formulations of traffic-flow problems. Focusing on the movement of individual vehicles in traffic, he first develops a model with instantaneous velocity control, expands that to velocity control with lag time, and finds an approximate solution for a short lag time. He establishes local and asymptotic stability of the solution, and then goes on to study the fluctuation of car spacings. The exercises ask the student to consider generalization of the results in the chapter and to investigate related models. The following chapter moves from investigating the movement of individual vehicles to questions of overall traffic flow, and so moves from Lagrangian to Eulerian perspectives.

Prerequisites are a solid background in calculus and ordinary differential equations. Parts of chapters 7 through 11 use partial differential equations and could be omitted in a course for less advanced students. As a whole the book is probably best suited for advanced undergraduates or beginning graduate students. 

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.