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Mathematical Modeling

Mark M. Meerschaert
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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Not too long ago there were only a handful of books designed to introduce students to mathematical modeling. Now there are many choices and a considerable diversity of approaches. The current book is the fourth edition of one of the more popular choices for introductory courses.

The author’s intention is to provide a general introduction for advanced undergraduate and beginning graduate students in mathematics or closely related fields. He assumes a standard undergraduate background including multivariable calculus, differential equations and linear algebra and suggests that some prior acquaintance with probability and statistics, although not necessary, would be useful.

This book distinguishes itself from comparable texts by its broad treatment of the field. It offers an extensive survey of mathematical modeling problems and techniques that is organized into three big sections corresponding to optimization, dynamics and probability models. The optimization section introduces basic modeling concepts and explores optimization in several contexts: single and multivariable, continuous and discrete, constrained and unconstrained, as well as linear and nonlinear programming.

In the realm of dynamic models the author discusses state space analysis for linear and nonlinear systems in continuous and discrete time, and then goes on to chaos and fractals, simulation, and numerical solutions of differential equations. Probability modeling includes discrete and continuous models, diffusions, Markov models, Monte Carlo simulation, and an introduction to time series analysis and linear regression.

Each of the big sections has a chapter on computation that addresses numerical methods. The author wants to encourage students to learn the use of appropriate technology for solving mathematical problems, and he actively uses software packages like Mathematica, Maple, Minitab, and the linear programming language LINDO. He provides pseudo-code for some algorithms, and his website gives access to the corresponding computer code for a variety of platforms.

The author makes it clear that the modeling process begins and ends in the world outside mathematics, and that the mathematical part in the middle is critical but not the be-all and end-all. He provides a lot of good examples, and he says that all his exercises come from real problems — none of them were made up.

One of my concerns about the book is the sheer number of topics that are treated in fairly rapid succession. Students are exposed to a considerable variety of modeling techniques, but the treatment is often too brief and sometimes borders on the superficial. Another concern is that, while the author’s five step approach to modeling is clear and direct, it is too simplistic. While he does address questions of sensitivity and robustness, he never quite grapples with the issues of model validation and sufficiency. Even robust and relatively insensitive models fail, often because of unmodeled effects. Validation is a very important part of good modeling practice.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1. One-Variable Optimization
2. Multivariable Optimization
3. Computational Methods for Optimization

4. Introduction to Dynamic Models
5. Analysis of Dynamic Models
6. Simulation of Dynamic Models

7. Introduction to Probability Models
8. Stochastic Models
9. Simulation of Probability Models