# Mathematical Modeling and Simulation: Introduction for Scientists and Engineers

# Mathematical Modeling and Simulation: Introduction for Scientists and Engineers

ISBN: 978-3-527-40758-3

Dec 2008

362 pages

## Description

This concise and clear introduction to the topic requires only basic knowledge of calculus and linear algebra - all other concepts and ideas are developed in the course of the book. Lucidly written so as to appeal to undergraduates and practitioners alike, it enables readers to set up simple mathematical models on their own and to interpret their results and those of others critically. To achieve this, many examples have been chosen from various fields, such as biology, ecology, economics, medicine, agricultural, chemical, electrical, mechanical and process engineering, which are subsequently discussed in detail.Based on the author`s modeling and simulation experience in science and engineering and as a consultant, the book answers such basic questions as: What is a mathematical model? What types of models do exist? Which model is appropriate for a particular problem? What are simulation, parameter estimation, and validation?

The book relies exclusively upon open-source software which is available to everybody free of charge. The entire book software - including 3D CFD and structural mechanics simulation software - can be used based on a free CAELinux-Live-DVD that is available in the Internet (works on most machines and operating systems).

Preface

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

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

"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. (...) This delightful book has two unbeatable features that should absolutely win the audience (...) First of all, it illuminates many important conceptual ideas of mathematical modelling (...) Second, (...) this book enthusiastically promotes open-source software that works on most computers and operating systems and is freely available on the web. (...) 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."

(Yuri V. Rogovchenko, Zentralblatt MATH, European Mathematical Society)

"The book is certainly a reference for those, beginners or professional, who search for a complete and easy to follow step-by-step guide in the amazing world of modeling and simulation (...) it is shown that mathematical models and simulation, if adequately used, help to reduce experimental costs by a better exploration of the information content of experimental data (...) it is explained how to analyze a real problem arising from science or engineering and how to best describe it through a mathematical model. A number of examples help the reader to follow step by step the basics of modelling."

(Marcello Vasta, Meccanica: International Journal of Theoretical and Applied Mechanics, Vol. 44(3), 2009)

"The broad subject area covered in this book reflects the background of the author, an experienced mathematical consultant and academic (...) This book differs from almost all other available modeling books in that the author addresses both mechanistic and statistical models as well as "hybrid" models. Since many problems coming out of industrial and medical applications in recent years require hybrid models, this text is timely. The modeling range is enormous (...) In this single chapter ("Phenomenological Models") he manages to cover almost all the material one would expect to find in an undergraduate statistics program. (...) Parameter sensitivity and overfitting problems are discussed in a very simple context - very nice! (...) The author points out that, by translating a real-world problem into a mathematical form, one brings to bear on that problem the vast knowledge and powerful and free software tools available within the "mathematical universe", and his aim is to enable the reader to source this information. (...) I believe the author has succeeded in providing access to the available tools and an understanding of how to go about using these tools to solve real-world problems."

Neville Fowkes (University of Western Australia) in: SIAM Rev. 53(2), 2011, pp. 387-388 (Society of Industrial and Applied Mathematics, Philadelphia, USA)

(Yuri V. Rogovchenko, Zentralblatt MATH, European Mathematical Society)

"The book is certainly a reference for those, beginners or professional, who search for a complete and easy to follow step-by-step guide in the amazing world of modeling and simulation (...) it is shown that mathematical models and simulation, if adequately used, help to reduce experimental costs by a better exploration of the information content of experimental data (...) it is explained how to analyze a real problem arising from science or engineering and how to best describe it through a mathematical model. A number of examples help the reader to follow step by step the basics of modelling."

(Marcello Vasta, Meccanica: International Journal of Theoretical and Applied Mechanics, Vol. 44(3), 2009)

"The broad subject area covered in this book reflects the background of the author, an experienced mathematical consultant and academic (...) This book differs from almost all other available modeling books in that the author addresses both mechanistic and statistical models as well as "hybrid" models. Since many problems coming out of industrial and medical applications in recent years require hybrid models, this text is timely. The modeling range is enormous (...) In this single chapter ("Phenomenological Models") he manages to cover almost all the material one would expect to find in an undergraduate statistics program. (...) Parameter sensitivity and overfitting problems are discussed in a very simple context - very nice! (...) The author points out that, by translating a real-world problem into a mathematical form, one brings to bear on that problem the vast knowledge and powerful and free software tools available within the "mathematical universe", and his aim is to enable the reader to source this information. (...) I believe the author has succeeded in providing access to the available tools and an understanding of how to go about using these tools to solve real-world problems."

Neville Fowkes (University of Western Australia) in: SIAM Rev. 53(2), 2011, pp. 387-388 (Society of Industrial and Applied Mathematics, Philadelphia, USA)