Methods of Mathematical Modelling

Many problems in the applied mathematical sciences involve at least three principal components: modeling (or modelling), analysis, and simulation. In research, an individual will often work within only a strict subset of these components. Moreover, every applied mathematician has their favorite techniques or specializes in a particular type of modeling, analysis, or simulation. A difficult problem for the applied mathematics community and educators in applied mathematics then becomes: what are the modeling, analysis and simulation techniques that every student of applied mathematics, particularly at the undergraduate level, should be exposed to? Furthermore, to what extent do the applied mathematics educational needs of mathematics students correspond to the applied mathematics educational needs of students in other sciences or engineering?

The book Methods of Mathematical Modelling, Continuous Systems and Differential Equations, here abbreviated as MoMM, by Thomas Witelski and Mark Bowen, contains material that one could easily argue belongs on the list of essential modeling and analysis techniques that every student of applied mathematics should be exposed to at a relatively early stage. In fact, much of what is in this book is also extremely valuable for students in science or engineering. The book is written for undergraduate students and is a recent addition to the Springer Undergraduate Mathematics Series (SUMS).

The main focus of MoMM is physical modeling with differential equations, both ordinary and partial. Five of the twelve chapters are concerned with modeling techniques, while the remaining seven develop pencil and paper, i.e., analytic, techniques for analyzing or approximating solutions to the differential equations that arise as mathematical models. I found chapters 4, on dimensional analysis, 6, on perturbation methods, and 10, on fast/slow systems, particularly enjoyable.

The text is well written. The authors have provided a clear and concise presentation of many important topics in a way that should be accessible to students following a first course in differential equations. Some exposure to partial differential equations and a knowledge of physics at the level of a first calculus-based course would probably be beneficial, but is not essential. More advanced students could easily learn a significant amount of useful mathematics reading the text independently. Many carefully chosen examples and illustrations aid in digesting the techniques introduced. There is an ample supply of end of chapter exercises for students to test their understanding and gain experience in applying techniques. Solutions to some of the exercises are provided at the end of the text. Also, the book includes a list of references that a student could use to find sources for further study of the material introduced in MoMM.

It is appropriate to mention that there is considerable overlap between MoMM and other texts, such as Applied Mathematics by Logan and Introduction to the Foundations of Applied Mathematics by Holmes. The books of Logan and Holmes are both listed in the references in MoMM. Principles of Applied Mathematics by Keener could make for an interesting followup text for students who enjoy the content of MoMM. While there is some overlap between Keener’s text and MoMM, Principles of Applied Mathematics contains some techniques from functional analysis, making it a slightly more sophisticated text.

As a personal judgement, I would be less inclined to use MoMM as a text for a course with mathematical modeling in the title, but more for a course called something like techniques in applied math. This is because I feel a course in mathematical modeling should introduce a broader range of modeling techniques and include connections between models and data, which are not discussed in MoMM. Furthermore, if I were to use MoMM as a course text, which I certainly would like to do, I would probably not try to cover content from each of the twelve chapters. Instead, I would select six to eight of the chapters and then go slightly deeper than the text does into the areas represented by the chosen chapters. Most of the chapters are ten to twenty pages in total, including exercises which I feel does not allow for as full a coverage of some of the topics as may be desired. Nevertheless, in my opinion, Methods of Mathematical Modelling is a welcome addition to the SUMS series and should prove to be useful for many instructors and students.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.