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Mathematical modeling and simulation. Introduction for scientists and engineers. (English) Zbl 1156.00014
Weinheim: Wiley-VCH (ISBN 978-3-527-40758-3/pbk; 978-3-527-62760-8/ebook). xiv, 348 p. (2009).
This nicely and thoroughly written introductory text on mathematical modeling and simulation has been designed primarily for undergraduate students in engineering and science. Actually, the text addresses a much wider audience developing the author’s idea that “everyone models and simulates.” Starting with elementary examples and reasoning, Professor Velten suggests an elegant approach to mathematical modeling carefully going through all important steps from identification of a problem, definition of the associated system under study and analysis of the system’s properties to design of a mathematical model for the system, its numerical simulation and validation. Surprisingly, the text contains not so much mathematics, yet enough to maintain the rigor of presentation without scaring not-so-mathematically oriented readers, and the only mathematical prerequisites are standard calculus and linear algebra courses. The exposition is self-contained and main ideas are developed in the course of the book.
This delightful book has two unbeatable features that should absolutely win the audience. First of all, it illuminates many important conceptual ideas of mathematical modeling, often taken no notice in contemporary texts on the subject, through the author’s philosophical reflections and extensive experience in modeling. Second, contrary to so many texts on mathematical modeling and simulation based on the use of commercial software like MATLAB, Maple, or Mathematica, this book enthusiastically promotes open-source software (CAELinux-Live-DVD) that works on most computers and operating systems and is freely available on the web.
The material is arranged into the four chapters. Chapter 1 describes principles of mathematical modeling, emphasizes the role of mathematics as a natural language for modeling, introduces the concept of a mathematical model and provides a nice and detailed classification of mathematical models. Phenomenological models (i.e. models constructed only on experimental data without referring to a priori information about the system) are addressed in detail in Chapter 2. The topics discussed cover some elementary statistics, linear, multiple linear and nonlinear regression, neural networks, design of experiments and some modern phenomenological modeling approaches like soft computing, discrete event simulation, or signal processing. Mechanistic models (i.e. models constructed using some a priori information about the system) are studied in Chapters 3 and 4, where the crucial role of ordinary and partial differential equations in mathematical modeling is emphasized. In Chapter 3 the reader learns how ODEs arise in mathematical modeling and how one can set up models based on ODEs along with the basic theory of ODEs, closed form and numerical solutions of ODEs. Special attention is paid to fitting ODEs to data, which is crucial for validation and adjustment of a model. Nice selection of examples from population biology, wine fermentation and pharmacokinetics helps to illustrate main ideas and techniques. Chapter 4 starts with an explanation of limitations of ODEs that require use of more complicated PDEs for modeling spatially nonhomogeneous phenomena. Some basic theoretical facts regarding PDEs are provided and techniques for obtaining closed form solutions are discussed. The finite-difference and finite-element methods are presented as most popular numerical methods for solving PDEs and the choice of a finite-element software is also addressed. As in Chapter 3, a number of interesting examples including, for instance, PDEs describing flow in porous media, diffusion and convention are considered. The chapter concludes with a brief overview of other mechanistic modeling approaches (difference equations, cellular automata, optimal control problems, differential-algebraic problems and inverse problems). Three appendices collect brief information on the free software package CAELinux, programming language and software environment R, and Maxima software. Finally, the reader will find an exhaustive list of almost two hundred and fifty useful references and a well-designed and functional index.
Very solid introductory text at the undergraduate level aimed at wide audience. Perfectly fits introductory modeling courses at colleges and universities that prefer to use open-source software rather than commercial one, and is an enjoyable reading in the first place. Highly recommended both as a main text and a supplementary one.

MSC:
00A71 General theory of mathematical modeling
00A69 General applied mathematics
00A72 General theory of simulation
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