Mathematical Modeling and Simulation : Introduction for Scientists and Engineers

• Autore: Velten Kai
• Pubblicazione: Newark : , : John Wiley & Sons, Incorporated, , 2024; ©2024
• Edizione: 2nd ed.
• Descrizione fisica: 1 online resource (499 pages)
• Disciplina: 511.8
• Soggetto topico: Mathematical models
• Altri autori: SchmidtDominik M KahlenKatrin
• Nota di contenuto: Cover -- Title Page -- Copyright -- Contents -- Preface -- Chapter 1 Principles of Mathematical Modeling -- 1.1 A Complex World Needs Models -- 1.2 Systems, Models, Simulations -- 1.2.1 Teleological Nature of Modeling and Simulation -- 1.2.2 Modeling and Simulation Scheme -- 1.2.3 Simulation -- 1.2.4 System -- 1.2.5 Conceptual and Physical Models -- 1.3 Mathematics as a Natural Modeling Language -- 1.3.1 Input-Output Systems -- 1.3.2 General Form of Experimental Data -- 1.3.3 Distinguished Role of Numerical Data -- 1.4 Definition of Mathematical Models -- 1.5 Examples and Some More Definitions -- 1.5.1 State Variables and System Parameters -- 1.5.2 Using Computer Algebra Software -- 1.5.3 The Problem‐Solving Scheme -- 1.5.4 Strategies to Set Up Simple Models -- 1.5.4.1 Mixture Problem -- 1.5.4.2 Tank Labeling Problem -- 1.5.4.3 Financial Mathematics -- 1.5.5 Linear Programming -- 1.5.6 Modeling a Black Box System -- 1.6 Even More Definitions -- 1.6.1 Phenomenological and Mechanistic Models -- 1.6.2 Stationary and Instationary Models -- 1.6.3 Distributed and Lumped Models -- 1.7 Classification of Mathematical Models -- 1.7.1 From Black to White Box Models -- 1.7.2 SQM Space Classification: S Axis -- 1.7.3 SQM Space Classification: Q Axis -- 1.7.4 SQM Space Classification: M Axis -- 1.8 Everything Looks Like a Nail? -- Chapter 2 Phenomenological Models -- 2.1 Elementary Statistics -- 2.1.1 Descriptive Statistics -- 2.1.1.1 Using Calc or Excel -- 2.1.1.2 Using R in RStudio -- 2.1.1.3 Roadmap for a First Analysis -- 2.1.2 Random Processes and Probability -- 2.1.2.1 Random Variables -- 2.1.2.2 Probability -- 2.1.2.3 Densities and Distributions -- 2.1.2.4 The Uniform Distribution -- 2.1.2.5 The Normal Distribution -- 2.1.2.6 Expected Value and Standard Deviation -- 2.1.2.7 More on Distributions -- 2.1.2.8 Quantiles and Confidence Intervals. ; 2.1.3 Inferential Statistics -- 2.1.3.1 Is Crop A's Yield Really Higher? -- 2.1.3.2 Structure of a Hypothesis Test -- 2.1.3.3 The t‐test -- 2.1.3.4 Testing Normality -- 2.1.3.5 Type I/II Errors, Power, and Effect Size -- 2.1.3.6 Testing Regression Parameters -- 2.1.3.7 Analysis of Variance -- 2.2 Linear Regression -- 2.2.1 The Linear Regression Problem -- 2.2.2 Solution Using Software -- 2.2.3 The Coefficient of Determination -- 2.2.4 Interpretation of the Regression Coefficients -- 2.2.5 Checking Assumptions -- 2.2.6 Nonlinear Linear Regression -- 2.3 Multiple Linear Regression -- 2.3.1 The Multiple Linear Regression Problem -- 2.3.2 Solution Using Software -- 2.3.3 Cross‐Validation -- 2.4 Nonlinear Regression -- 2.4.1 The Nonlinear Regression Problem -- 2.4.2 Solution Using Software -- 2.4.3 Multiple Nonlinear Regression -- 2.4.4 Implicit and Vector‐Valued Problems -- 2.5 Smoothing Splines -- 2.6 Neural Networks -- 2.6.1 General Idea -- 2.6.2 Feed‐Forward Neural Networks -- 2.6.3 Solution Using Software -- 2.6.4 Interpretation of the Results -- 2.6.5 Generalization and Overfitting -- 2.6.6 Several Inputs Example -- 2.7 Big Data Analysis -- 2.7.1 From Data to Knowledge -- 2.7.2 Artificial Data -- 2.7.3 Influencing Factors and Interactions for z1 -- 2.7.4 Influencing Factors and Interactions for z2 and z3 -- 2.7.5 Dimensional Reduction and Classification -- 2.7.5.1 Principal Component Analysis, Factor Analysis, and Correspondence Analysis -- 2.7.5.2 Classification -- 2.7.6 Conclusions -- 2.8 Signal Processing -- 2.8.1 Example, Idea, and Useful R Packages -- 2.8.2 Time‐Series Classification Using tsfresh, Python and R -- 2.9 Design of Experiments -- 2.9.1 Completely Randomized Design -- 2.9.2 Randomized Complete Block Design -- 2.9.3 Latin Square Design -- 2.9.4 Factorial Designs -- 2.9.5 Optimal Sample Size -- 2.9.6 DOE Workflow. ; 2.9.7 Optimal Designs -- 2.10 Other Phenomenological Modeling Approaches -- 2.10.1 Soft Computing -- 2.10.1.1 Fuzzy Model of a Washing Machine -- 2.10.2 Discrete Event Simulation -- Chapter 3 Mechanistic Models I: ODEs -- 3.1 Distinguished Role of Differential Equations -- 3.2 Introductory Examples -- 3.2.1 Archaeology Analogy -- 3.2.2 Body Temperature -- 3.2.2.1 Phenomenological Model -- 3.2.2.2 Application -- 3.2.3 Alarm Clock -- 3.2.3.1 Need for a Mechanistic Model -- 3.2.3.2 Applying the Modeling and Simulation Scheme -- 3.2.3.3 Setting Up the Equations -- 3.2.3.4 Comparing Model and Data -- 3.2.3.5 Validation Fails - What Now? -- 3.2.3.6 A Different Way to Explain the Temperature Memory -- 3.2.3.7 Limitations of the Model -- 3.3 General Idea of ODE's -- 3.3.1 Intrinsic Meaning of pi -- 3.3.2 ex Solves an ODE -- 3.3.3 Infinitely Many Degrees of Freedom -- 3.3.4 Intrinsic Meaning of the Exponential Function -- 3.3.5 ODEs as a Function Generator -- 3.4 Setting Up ODE Models -- 3.4.1 Body Temperature Example -- 3.4.1.1 Formulation of an ODE Model -- 3.4.1.2 ODE Reveals the Mechanism -- 3.4.1.3 ODE's Connect Data and Theory -- 3.4.1.4 Three Ways to Set Up ODEs -- 3.4.2 Alarm Clock Example -- 3.4.2.1 A System of Two ODEs -- 3.4.2.2 Parameter Values Based on A Priori Information -- 3.4.2.3 Result of a Hand‐Fit -- 3.4.2.4 A Look into the Black Box -- 3.5 Some Theory You Should Know -- 3.5.1 Basic Concepts -- 3.5.2 First‐Order ODEs -- 3.5.3 Autonomous, Implicit, and Explicit ODEs -- 3.5.4 The Initial Value Problem -- 3.5.5 Boundary Value Problems -- 3.5.6 Example of Nonuniqueness -- 3.5.7 ODE Systems -- 3.5.8 Linear Versus Nonlinear -- 3.6 Solution of ODE's: Overview -- 3.6.1 Toward the Limits of Your Patience -- 3.6.2 Closed Form Versus Numerical Solutions -- 3.7 Closed Form Solutions -- 3.7.1 Right‐Hand Side Independent of the Independent Variable. ; 3.7.1.1 General and Particular Solutions -- 3.7.1.2 Solution by Integration -- 3.7.1.3 Using Computer Algebra Software -- 3.7.1.4 Imposing Initial Conditions -- 3.7.2 Separation of Variables -- 3.7.2.1 Application to the Body Temperature Model -- 3.7.2.2 Solution Using Maxima and Mathematica -- 3.7.3 Variation of Constants -- 3.7.3.1 Application to the Body Temperature Model -- 3.7.3.2 Using Computer Algebra Software -- 3.7.3.3 Application to the Alarm Clock Model -- 3.7.3.4 Interpretation of the Result -- 3.7.4 Dust Particles in the ODE Universe -- 3.8 Numerical Solutions -- 3.8.1 Algorithms -- 3.8.1.1 The Euler Method -- 3.8.1.2 Example Application -- 3.8.1.3 Order of Convergence -- 3.8.1.4 Stiffness -- 3.8.2 Solving ODE's Using Maxima -- 3.8.2.1 Heuristic Error Control -- 3.8.2.2 ODE Systems -- 3.8.3 Solving ODEs Using R and lsoda -- 3.8.3.1 Local Error Control in lsoda -- 3.8.3.2 Effect of the Local Error Tolerances -- 3.8.3.3 A Rule of Thumb to Set the Tolerances -- 3.8.3.4 Example Applications -- 3.9 Fitting ODE's to Data -- 3.9.1 Parameter Estimation in the Alarm Clock Model -- 3.9.1.1 Estimating Two Parameters -- 3.9.1.2 Estimating Initial Values -- 3.9.1.3 Sensitivity of the Parameter Estimates -- 3.9.2 The General Parameter Estimation Problem -- 3.9.2.1 One State Variable Characterized by Data -- 3.9.2.2 Several State Variables Characterized by Data -- 3.9.3 Indirect Measurements Using Parameter Estimation -- 3.10 More Examples -- 3.10.1 Predator-Prey Interaction -- 3.10.1.1 Lotka-Volterra Model -- 3.10.1.2 General Dynamical Behavior -- 3.10.1.3 Nondimensionalization -- 3.10.1.4 Phase Plane Plots -- 3.10.2 Wine Fermentation -- 3.10.2.1 Setting Up a Mathematical Model -- 3.10.2.2 Yeast -- 3.10.2.3 Ethanol and Sugar -- 3.10.2.4 Nitrogen -- 3.10.2.5 Using a Hand‐Fit to Estimate N0 -- 3.10.2.6 Parameter Estimation. ; 3.10.2.7 Problems with Nonautonomous Models -- 3.10.2.8 Converting Data into a Function -- 3.10.2.9 Using Weighting Factors -- 3.10.3 Pharmacokinetics -- 3.10.4 Plant Growth -- Chapter 4 Mechanistic Models II: PDEs -- 4.1 Introduction -- 4.1.1 Limitations of ODE Models -- 4.1.2 Overview: Strange Animals, Sounds, and Smells -- 4.1.3 Two Problems You Should Be Able to Solve -- 4.2 The Heat Equation -- 4.2.1 Fourier's Law -- 4.2.2 Conservation of Energy -- 4.2.3 Heat Equation & -- equals -- Fourier's Law + Energy Conservation -- 4.2.4 Heat Equation in Multidimensions -- 4.2.5 Anisotropic Case -- 4.2.6 Understanding Off‐diagonal Conductivities -- 4.3 Some Theory You Should Know -- 4.3.1 Partial Differential Equations -- 4.3.1.1 First‐Order PDEs -- 4.3.1.2 Second‐Order PDEs -- 4.3.1.3 Linear Versus Nonlinear -- 4.3.1.4 Elliptic, Parabolic, and Hyperbolic Equations -- 4.3.2 Initial and Boundary Conditions -- 4.3.2.1 Well Posedness -- 4.3.2.2 A Rule of Thumb -- 4.3.2.3 Dirichlet and Neumann Conditions -- 4.3.3 Symmetry and Dimensionality -- 4.3.3.1 1D Example -- 4.3.3.2 2D Example -- 4.3.3.3 3D Example -- 4.3.3.4 Rotational Symmetry -- 4.3.3.5 Mirror Symmetry -- 4.3.3.6 Symmetry and Periodic Boundary Conditions -- 4.4 Closed‐Form Solutions -- 4.4.1 Problem 1 -- 4.4.2 Separation of Variables -- 4.4.3 A Particular Solution for Validation -- 4.5 Numerical Solution of PDEs -- 4.6 The Finite Difference Method -- 4.6.1 Replacing Derivatives with Finite Differences -- 4.6.2 Formulating an Algorithm -- 4.6.3 Implementation in R -- 4.6.4 Error and Stability Issues -- 4.6.5 Explicit and Implicit Schemes -- 4.6.6 Computing Electrostatic Potentials -- 4.6.7 Iterative Methods for the Linear Equations -- 4.6.8 Billions of Unknowns -- 4.7 The Finite Element Method -- 4.7.1 Weak Formulation of PDEs -- 4.7.2 Approximation of the Weak Formulation. ; 4.7.3 Appropriate Choice of the Basis Functions.
• Sommario/riassunto: This book, 'Mathematical Modeling and Simulation Introduction for Scientists and Engineers,' provides a comprehensive guide to mathematical modeling and simulation techniques used in science and engineering. It covers the principles of mathematical models, including system definitions, simulation techniques, and the application of differential equations. The book aims to equip scientists and engineers with the necessary tools to tackle complex problems by using mathematical models. With a focus on practical application, the text delves into statistical models, regression methods, and neural networks, offering examples from biology and ecology. The authors also introduce software tools such as OpenFOAM, Python, RStudio, and Maxima to facilitate the implementation of these models. The intended audience includes professionals and students in scientific and engineering fields who require a solid foundation in mathematical modeling.
• Titolo autorizzato: Mathematical Modeling and Simulation
• ISBN: 9783527839391; 3527839399
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9911020031703321
• Lo trovi qui: Univ. Federico II