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Mathematical Modeling and Simulation: Introduction for Scientists and Engineers

Kai Velten
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

An amazing number of books on mathematical modeling have appeared in the last several years. The book under review is one of the newest. It comes from a German professor and consultant whose background and experience are well-matched to the task. As the subtitle indicates, this book is aimed primarily toward undergraduates in the sciences and engineering. The author suggests that prerequisites include only some calculus and linear algebra. There is a heavy use of open software packages throughout; all the software is available on the Internet.

The book is divided into three main parts. The first is an extended introduction to the principles of modeling. The second part introduces phenomenological models (models based on data), and the third part focuses on mechanistic models (mostly models based on differential equations). The author takes up quite a large number of topics — he is clearly an enthusiastic teacher and can hardly restrain himself from including one new topic after another.

One valuable thing that books like this can offer is a solid introduction to the idea of modeling that describes goals and methods, and explains and distinguishes terms like “modeling” and “simulation”. In the first part of the book the author sets off in that direction. What results doesn’t really work. Consider the following selections:

A mathematical model is a triplet (S, Q, M) where S is a system, Q is a question relating to S, and M is a set of mathematical statements M = {Σ1, Σ2, ..., Σn} which can be used to answer Q. (page 12)

The system and the question relating to the system are indispensable parts of a mathematical model. It is a genuine property of mathematical models to be more mathematical than “l’art pour l’art”. (page 13)

The first selection is highlighted in the text as a “definition” and the second as a “note”, and they are representative of perhaps a dozen others. The first is unnecessarily formal, especially for the intended readers (and it appears already on page 12!), and it’s short on substance too. The second left me scratching my head; what is he really trying to say? Unfortunately, I found that much of the first part of the book reads like this. Lacking concrete examples, statements like these mean practically nothing to students.

The second and third parts of the book are better, but I still found them paradoxically either too wordy or too brief. The first problem is that too many topics are treated with too little detail. (Four pages each are given to inferential statistics and neural networks, for example.) Yet when the author does spend time on an example, he is often notably verbose. Nonetheless, there are a number of good examples. A simple example with a temperature sensor is used to investigate both phenomenological and mechanistic models, and to study adequacy and validity of the models. Another good example focuses on the enzyme kinetics of wine fermentation. This is a real example of moderate complexity with eighteen parameters that control the process.

The author is bursting with enthusiasm for the subject and clearly has broad experience and expertise to share with students. Unfortunately, it doesn’t come through well in his book. 

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. Principles of Mathematical Modeling. 

1.1 A complex world needs models.

1.2 Systems, models, simulations.

1.3 Mathematics is the natural modeling language.

1.4 Definition of mathematical models.

1.5 Examples and some more definitions.

1.6 Even more definitions.

1.7 Classification of mathematical models.

1.8 Everything looks like a nail?

2. Phenomenological models. 

2.1 Elementary statistics.

2.2 Linear regression.

2.3 Multiple linear regression.

2.4 Nonlinear regression.

2.5 Neural networks.

2.6 Design of experiments.

2.7 Other phenomenological modeling approaches.

3. Mechanistic models I: ODE's. 

3.1 Distinguished role of differential equations.

3.2 Introductory examples.

3.3 General idea of ODE's.

3.4 Setting up ODE models.

3.5 Some theory you should know.

3.6 Solution of ODE's: Overview.

3.7. Closed form solution.

3.8 Numerical solutions.

3.9 Fitting ODE's to data.

3.10 More examples.

4. Mechanistic models II: PDE's. 

4.1. Introduction.

4.2. The heat equation.

4.3. Some theory you should know.

4.4 Closed form solution.

4.5 Numerical solution of PDE's.

4.6 The finite difference method.

4.7 The finite element method.

4.8 Finite element software.

4.9 A sample session using Salome Meca.

4.10 A look beyond the heat equation.

4.11 Other mechanistic modeling approaches.

A CAELinux and the book software.

B R (programming language and software environment).

C Maxima.