Kai Velten is a mathematician and modeling and simulation consultant focusing on data analysis and differential equations. He held scientific positions at Braunschweig and Erlangen Universities and the Fraunhofer-ITWM in Kaiserslautern between 1990-2000, works as professor of mathematics at Hochschule Geisenheim University since 2000 and was offered another professorship at Lüneburg University in 2012.

Dominik M. Schmidt holds MSc (Beverage Technology) and PhD (Agriculture) degrees obtained from Hochschule Geisenheim University and University of Gießen. Working in the Department of Modeling and Systems Analysis of Hochschule Geisenheim University since 2013, he performed a broad range of research projects in the fields of data science, mathematical modeling and computational fluid dynamics.

Katrin Kahlen is a mathematician working at Hochschule Geisenheim University since 2011. She holds PhD (Mathematics) and habilitation (Agronomy) degrees obtained from University of Hannover and became Adjunct Professor at Hochschule Geisenheim University in 2021. She is particularly interested in modeling and simulation of plant-environment interactions, and the development and use of virtual plants.

Preface xv

1 Principles of Mathematical Modeling 1

1.1 A Complex World Needs Models 1

1.2 Systems, Models, Simulations 3

1.3 Mathematics as a Natural Modeling Language 8

1.4 Definition of Mathematical Models 10

1.5 Examples and Some More Definitions 11

1.6 Even More Definitions 32

1.7 Classification of Mathematical Models 36

1.8 Everything Looks Like a Nail? 41

2 Phenomenological Models 44

2.1 Elementary Statistics 45

2.2 Linear Regression 69

2.3 Multiple Linear Regression 79

2.4 Nonlinear Regression 87

2.5 Smoothing Splines 94

2.6 Neural Networks 97

2.7 Big Data Analysis 109

2.8 Signal Processing 119

2.9 Design of Experiments 124

2.10 Other Phenomenological Modeling Approaches 138

3 Mechanistic Models I: ODEs 142

3.1 Distinguished Role of Differential Equations 142

3.2 Introductory Examples 143

3.3 General Idea of ODE's 154

3.4 Setting Up ODE Models 158

3.5 Some Theory You Should Know 165

3.6 Solution of ODE's: Overview 174

3.7 Closed Form Solutions 176

3.8 Numerical Solutions 192

3.9 Fitting ODE's to Data 208

3.10 More Examples 216

4 Mechanistic Models II: PDEs 237

4.1 Introduction 237

4.2 The Heat Equation 241

4.3 Some Theory You Should Know 247

4.4 Closed-Form Solutions 259

4.5 Numerical Solution of PDEs 262

4.6 The Finite Difference Method 263

4.7 The Finite Element Method 269

4.8 The Finite Volume Method 276

4.9 Software Packages to Solve PDEs 283

4.10 A Sample Session on the Numerical Solution of Thermal Conduction 285

4.11 A Look Beyond the Heat Equation 316

4.12 Computational Fluid Dynamics (CFD) 328

4.13 Numerical Solutions of Example Flow Problems 331

4.14 Other Mechanistic Modeling Approaches 352

5 Virtualization 358

5.1 Introduction 358

5.2 Virtual Plants 359

5.3 Examples of Advanced Applications 371

6 Crashcourses and Book Software 395

6.1 Crashcourse R 395

6.2 Crashcourse Maxima 426

6.3 CrashCrashcourse Python (and all the rest) 433

6.4 Book Software and GmLinux 434

References 435

Index 455