Cover -- Title Page -- Copyright Page -- CONTENTS -- List of Figures -- Preface -- Acknowledgments -- Editor and Contributors -- PART I INTRODUCTION AND FOUNDATIONS -- 1 Differential Equations -- 1.1 Ordinary Differential Equations -- 1.1.1 First-Order ODEs -- 1.1.2 Higher-Order ODEs -- 1.1.3 Linear System -- 1.1.4 Sturm-Liouville Equation -- 1.2 Partial Differential Equations -- 1.2.1 First-Order PDEs -- 1.2.2 Classification of Second-Order PDEs -- 1.3 Classic Mathematical Models -- 1.4 Other Mathematical Models -- 1.5 Solution Techniques -- 1.5.1 Separation of Variables -- 1.5.2 Laplace Transform -- 1.5.3 Similarity Solution -- 1.5.4 Change of Variables -- Exercises -- 2 Mathematical Modeling -- 2.1 Mathematical Modeling -- 2.2 Model Formulation -- 2.3 Parameter Estimation -- 2.4 Mathematical Models -- 2.4.1 Differential Equations -- 2.4.2 Functional and Integral Equations -- 2.4.3 Statistical Models -- 2.4.4 Rule-based Models -- 2.5 Numerical Methods -- 2.5.1 Numerical Integration -- 2.5.2 Numerical Solutions of PDEs -- Exercises -- 3 Numerical Methods: An Introduction -- 3.1 Direct Integration -- 3.1.1 Euler Scheme -- 3.1.2 Leap-Frog Method -- 3.1.3 Runge-Kutta Method -- 3.2 Finite Difference Methods -- 3.2.1 Hyperbolic Equations -- 3.2.2 Second-Order Wave Equation -- 3.2.3 Parabolic Equation -- 3.2.4 Elliptical Equation -- Exercises -- 4 Teaching Mathematical Modeling in Teacher Education: Efforts and Results -- 4.1 Introduction -- 4.2 Theoretical Frameworks Connected to Mathematical Modeling -- 4.2.1 Instrumental Competence -- 4.2.2 The Importance of Variation -- 4.3 Mathematical Modeling Tasks -- 4.4 Conclusions -- Exercises -- PART II MATHEMATICAL MODELING WITH MULTIDISCIPLINARY APPLICATIONS -- 5 Industrial Mathematics with Applications -- 5.1 Industrial Mathematics -- 5.2 Numerical Simulation of Metallurgical Electrodes.

5.2.1 The Industrial Problem: Metallurgy of Silicon -- 5.2.2 Mathematical Modeling -- 5.2.3 Numerical Solution -- 5.2.4 Numerical Results -- 5.3 Numerical Simulation of Pit Lake Water Quality -- 5.3.1 Introduction to the Problem -- 5.3.2 A Stirred Tank Model to Predict Pit Lake Water Quality -- 5.3.3 Mathematical Models for Chemical Reaction Systems -- 5.3.4 Numerical Solution of the Model -- 5.3.5 Numerical Results: A Simplified Chemical Problem. -- Exercises -- 6 Binary and Ordinal Data Analysis in Economics: Modeling and Estimation -- 6.1 Introduction -- 6.2 Theoretical Foundations -- 6.2.1 Binary Outcomes -- 6.2.2 Ordinal Outcomes -- 6.3 Estimation -- 6.3.1 Maximum Likelihood Estimation -- 6.3.2 Bayesian Estimation -- 6.3.3 Marginal Effects -- 6.4 Applications -- 6.4.1 Women's Labor Force Participation -- 6.4.2 An Ordinal Model of Educational Attainment -- 6.5 Conclusions -- Exercises -- 7 Inverse Problems in ODEs -- 7.1 Banach's Fixed Point Theorem & The Collage Theorem -- 7.2 Existence-Uniqueness of Solutions to Initial Value Problems -- 7.3 Solving Inverse Problems for ODEs -- Exercises -- References -- 8 Estimation of Model Parameters -- 8.1 Estimation is an Inverse Problem -- 8.2 The Multivariate Normal Distribution -- 8.3 Model of Observations -- 8.3.1 Deterministic Model and its Linearization -- 8.3.2 Probabilistic Model -- 8.4 Estimation -- 8.4.1 Bayesian Inference -- 8.4.2 Moment Matching -- 8.4.3 Estimation by Optimization -- 8.5 Conclusion -- Exercises -- 9 Linear and Nonlinear Parabolic Partial Differential Equations in Financial Engineering -- 9.1 Financial Derivatives -- 9.2 Motivation for a Model for the Price of Stocks -- 9.3 Stock Prices Involving the Wiener Process -- 9.4 Connection Between the Wiener Process and PDEs -- 9.5 The Black-Scholes-Merton Equation -- 9.6 Solution of the Black-Scholes-Merton Equation.

9.7 Free Boundary-Value Problems -- 9.8 The Hamilton-Jacobi-Bellman Equation -- 9.8.1 The Hamilton-Jacobi-Bellman Equation -- 9.8.2 An Explicitly Worked Example -- 9.8.3 Viscosity Solutions -- 9.9 Numerical Methods -- 9.9.1 The Crank-Nicholson Method -- 9.9.2 Numerical Treatment of Variational Inequalities -- 9.9.3 Numerical Treatment of HJB Equations -- 9.10 Conclusion -- Exercises -- 10 Decision Modeling in Supply Chain Management -- 10.1 Introduction to Decision Modeling -- 10.1.1 The Origin of Decision Modeling -- 10.1.2 Definition of Decision Modeling -- 10.1.3 Data in Decision Modeling -- 10.1.4 Role of Spreadsheets in Decision Modeling -- 10.1.5 Types of Decision Models -- 10.1.6 Steps of Decision Modeling -- 10.2 Mathematical Programming Models -- 10.2.1 Introduction of Linear Programming Models -- 10.2.2 Properties of a Linear Programming Model -- 10.2.3 Assumptions of a Linear Programming Model -- 10.2.4 Other Mathematical Programming Models -- 10.3 Introduction of Supply Chain Management -- 10.3.1 Importance of Supply Chain Management -- 10.3.2 Activities in Supply Chain Management -- 10.4 Applications in Supply Chain Management -- 10.4.1 Manufacturing Applications -- 10.4.2 Transportation Applications -- 10.4.3 Assignment Applications -- 10.5 Summary -- Exercises -- 11 Modeling Temperature for Pricing Weather Derivatives -- 11.1 Introduction -- 11.2 Stochastic Temperature Modeling -- 11.2.1 Simple Stochastic Mean Reverting Processes -- 11.3 Continuous-Time Autoregressive Processes -- 11.3.1 An Empirical Study -- 11.4 Pricing of Temperature Futures Contracts -- Exercises -- 12 Decision Theory under Risk and Applications in Social Sciences: I. Individual Decision Making -- 12.1 Introduction -- 12.2 The Fundamental Framework -- 12.3 A Brief Introduction to Theory of Choice -- 12.4 Collective Choice -- 12.5 Preferences Under Uncertainty.

12.6 Decisions Over Time -- 12.7 The Problem of Aggregation -- 12.7.1 Aggregation of Time Preferences -- 12.7.2 Aggregation of Beliefs -- 12.8 Conclusion -- Exercises -- 13 Fractals, with Applications to Signal and Image Modeling -- 13.1 Iterated Function Systems -- 13.2 Fractal Dimension -- 13.3 More on the Definition of Iterated Function System -- 13.4 The Chaos Game -- 13.5 An Application to Image Analysis -- References -- 14 Efficient Numerical Methods for Singularly Perturbed Differential Equations -- 14.1 Introduction -- 14.2 Characterization of SPPs -- 14.3 Numerical Approximate Solution -- 14.3.1 Failure of Classical Finite Difference Schemes on Uniform Meshes -- 14.3.2 Exponentially Fitted Difference Scheme -- 14.4 SPPs Arising in Chemical Reactor Theory -- 14.4.1 Initial-Value Technique -- 14.4.2 Boundary-Value Technique -- 14.4.3 Shooting Method -- 14.4.4 Booster Method -- 14.4.5 Semilinear Problems -- 14.5 Layer-Adapted Nonuniform Meshes -- 14.5.1 Bakhvalov Meshes -- 14.5.2 Shishkin Meshes -- 14.5.3 Equidistribution Meshes -- PART III ADVANCED MODELING TOPICS -- 15 Fractional Calculus and its Applications -- 15.1 Introduction -- 15.2 Fractional Calculus Fundamentals -- 15.2.1 Special Functions -- 15.2.2 Definitions of Fractional Operator -- 15.2.3 Grünwald-Letnikov Fractional Derivatives -- 15.2.4 Riemann-Liouville Fractional Derivatives -- 15.2.5 Caputo Fractional Derivatives -- 15.2.6 Laplace Transform Method -- 15.2.7 Some Properties of Fractional Calculus -- 15.2.8 Numerical Methods for Fractional Calculus -- 15.3 Fractional-Order Systems and Controllers -- 15.3.1 Fractional LTI Systems -- 15.3.2 Fractional Nonlinear Systems -- 15.3.3 Fractional-Order Controllers -- 15.4 Stability of Fractional-Order Systems -- 15.4.1 Stability of Fractional LTI Systems -- 15.4.2 Stability of Fractional Nonlinear Systems.

15.5 Applications of Fractional Calculus -- 15.5.1 Control of Electrical Heater -- 15.5.2 Memristor-Based Chua's Circuit -- 15.5.3 Viscoelastic Models of Cells -- Exercises -- 16 The Goal Programming Model: Theory and Applications -- 16.1 Multi-Criteria Decision Aid -- 16.2 The Goal Programming Model -- 16.3 Scenario-based Goal Programming -- 16.4 Applications -- 16.4.1 A Goal Programming Model for Portfolio Selection -- 16.4.2 A Goal Programming Model for Media Management and Planning -- 16.4.3 A Goal Programming Model for Site Selection -- 16.4.4 A Goal Programming Model for the Next Release Problem -- Exercises -- 17 Decision Theory under Risk and Applications in Social Sciences: II Game Theory -- 17.1 Introduction -- 17.2 Best Replies and Nash Equilibria -- 17.3 Mixed Strategies and Minimax -- 17.4 Nash Equilibria and Conservative Strategies -- 17.5 Zero-Sum Games and the Minimax Theorem -- 17.6 Nash Equilibria for Mixed Strategies -- 17.7 Cooperative Games -- 17.8 Conclusion -- Exercises -- 18 Control Problems on Differential Equations -- 18.1 Introduction -- 18.2 Ordinary Differential Equations -- 18.2.1 Model Formulation -- 18.2.2 Controllability -- 18.2.3 Kalman's Rank Condition -- 18.3 Partial Differential Equations -- 18.3.1 Model Formulation -- 18.3.2 Controllability -- 18.3.3 Adjoint System and Observability -- Exercises -- 19 Markov-Jump Stochastic Models for Tropical Convection -- 19.1 Introduction -- 19.2 Random Numbers: Theory and Simulations -- 19.2.1 Random Variables -- 19.2.2 Mean, Variance, and Expectation -- 19.2.3 Conditional Probability -- 19.2.4 Law of Large Numbers -- 19.2.5 Monte Carlo Integration -- 19.2.6 Inverse Transform Method -- 19.2.7 Acceptance-Rejection Method -- 19.3 Markov Chains and Birth-Death Processes -- 19.3.1 Discrete-Time Markov Chains -- 19.3.2 The Poisson Process -- 19.3.3 Continuous-Time Markov Chains.

19.4 A Birth-Death Process for Convective Inhibition.