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Comprehensive Applied Mathematical Modeling in the Natural and Engineering Sciences

David J. Wolkind and Bonni J. Dichone
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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This ambitious textbook on mathematical modeling is intended for use as a text in a two-semester course for graduate students or advanced undergraduates. It assumes a background of advanced calculus and differential equations as well as basic understanding of the general principles of physical and biological science. The book is distinguished from other works in its category both by the number and kinds of case studies that it considers and by the authors’ intention to be comprehensive.

Comprehensive applied mathematical modeling has a specific meaning to the authors and it governs the way that they have organized their book. In general terms its goal is to clarify and describe scientific concepts and phenomena using mathematics. More specifically, it means following a sequence of formulating a scientific problem in mathematical terms, solving the resulting mathematical problem, and then interpreting and evaluating the results.

This makes the process sound more cut-and-dried than it ever is in practice. It is often an iterative effort that requires many passes. The authors are clearly aware of this, and they attend to issues of reducing models to simplified forms that retain important features of the original systems while achieving tractable mathematical problems.

The authors begin with a prototype problem intended to illustrate their approach throughout the book. They consider how to determine the escape velocity of a projectile under different assumptions on the form of the gravitational force. In doing so they introduce the idea of using non-dimensional variables and regular perturbation theory.

The book is divided into four parts with case studies and mathematical methods of increasing complexity. The first part focuses on modeling phenomena related to nonmoving continua. The applications include a predator-prey model with mites on fruit trees, heat conduction on finite and infinite bars, slime mold aggregation, soap bubbles and more. Techniques of linear stability theory are the focus here.

Part 2 shifts the emphasis to modeling moving continua. The applications are largely in fluid mechanics beginning with simple flows and moving on to inviscid and viscous flow problems. Part 3 describes the application on nonlinear stability theory in four case studies. These include modeling the behavior of laser light in an optical ring cavity and investigating the formation of rhombic patterns of vegetation in arid regions.

Part 4 has a miscellaneous collection of discussions of mathematical methods not otherwise treated in the book. These include applying the calculus of variations in constrained optimization and using Laplace and Fourier transform methods to solve wave and heat equations. This last part also includes a short chapter on finite mathematical models, and then concludes with six capstone problems.

“Pastoral interludes” are important elements of the text. These are present in many of the chapters. Their purpose is to introduce mathematical methods that are used in the associated case studies and thereby make the book as self-contained as possible. These interludes — there are more than twenty — cover topics like regular perturbation theory, stability analysis, calculus of variations, asymptotic series, Bessel functions, and more. They are woven into the text and well integrated. (The adjective “pastoral” seems a little unusual in this context; perhaps it is meant to suggest a gentle sort of guidance.)

In most respects this is a very attractive text with well-developed case studies. Readers are exposed to a substantial collection of powerful modeling tools. The case studies focus heavily on continuum mechanics, and finite models are treated only briefly. One might have wished for a broader reach, but the book is quite lengthy as it is. Where the authors get into the details of mathematics and science, the exposition is smooth and the writing clear and direct; this is less true in the general and introductory sections where the writing often seems stiff.

Each chapter has an abundance of well-chosen exercises. The capstone problems are also appealing and suggest excellent end-of-term projects.

Bill Satzer ([email protected]) 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.

See the table of contents in the publisher's webpage.