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Mathematical Modeling: Models, Analysis and Applications

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Mathematical Modeling: Models, Analysis and Applications, Second Edition introduces models of both discrete and continuous systems. This book is aimed at newcomers who desires to learn mathematical modeling, especially students taking a first course in the subject. Beginning with the step-by-step guidance of model formulation, this book equips the reader about modeling with difference equations (discrete models), ODE’s, PDE’s, delay and stochastic differential equations (continuous models). This book provides interdisciplinary and integrative overview of mathematical modeling, making it a complete textbook for a wide audience.

A unique feature of the book is the breadth of coverage of different examples on mathematical modelling, which include population models, economic models, arms race models, combat models, learning model, alcohol dynamics model, carbon dating, drug distribution models, mechanical oscillation models, epidemic models, tumor models, traffic flow models, crime flow models, spatial models, football team performance model, breathing model, two neuron system model, zombie model and model on love affairs. Common themes such as equilibrium points, stability, phase plane analysis, bifurcations, limit cycles, period doubling and chaos run through several chapters and their interpretations in the context of the model have been highlighted. In chapter 3, a section on estimation of system parameters with real life data for model validation has also been discussed.

Features

    • Covers discrete, continuous, spatial, delayed and stochastic models.

    • Over 250 illustrations, 300 examples and exercises with complete solutions.

    • Incorporates MATHEMATICA® and MATLAB®, each chapter contains Mathematica and Matlab codes used to display numerical results (available at CRC website).

    • Separate sections for Projects. Several exercise problems can also be used for projects.

    • Presents real life examples of discrete and continuous scenarios.

    The book is ideal for an introductory course for undergraduate and graduate students, engineers, applied mathematicians and researchers working in various areas of natural and applied sciences.

    Author(s): Sandip Banerjee
    Edition: 2
    Publisher: Chapman and Hall/CRC
    Year: 2021

    Language: English
    Commentary: Publisher PDF
    Pages: 434
    City: Boca Raton, FL
    Tags: Discrete Models; Difference Equations; Spatial Models

    Cover
    Half Title
    Title Page
    Copyright Page
    Dedication
    Contents
    Foreword
    Preface to Second Edition
    Author
    1. About Mathematical Modeling
    1.1. What Is Mathematical Modeling?
    1.2. History of Mathematical Modeling
    1.3. Importance of Mathematical Modeling
    1.4. Latest Developments in Mathematical Modeling
    1.5. Limitations of Mathematical Modeling
    1.6. Units
    1.7. Dimensions
    1.8. Dimensional Analysis
    1.9. Scaling
    1.10. How to Built Mathematical Models
    1.10.1. Step I (The Start)
    1.10.2. Step II (The Assumption)
    1.10.3. Step III (Schematic or Flow Diagrams)
    1.10.4. Step IV (Choosing Mathematical Equations)
    1.10.5. Step V (Solving Equations)
    1.10.6. Step VI (Interpretation of the Result)
    1.11. Mathematical Models and Functions
    1.11.1. Linear Models
    1.11.2. Quadratic Models
    1.11.3. Cubic Models
    1.11.4. Logistic Function and Logistic Growth Model
    1.11.5. Gompertz Function and Gompertz Growth Model
    1.12. Functional Responses in Population Dynamics
    1.12.1. Holling Type I Functional Response
    1.12.2. Holling Type II Functional Response
    1.12.3. Holling Type III Functional Response
    1.13. Miscellaneous Examples
    1.14. Exercises
    2. Discrete Models Using Difference Equations
    2.1. Difference Equations
    2.1.1. Linear Difference Equation with Constant Coefficients
    2.1.2. Solution of Homogeneous Equations
    2.1.3. Difference Equations: Equilibria and Stability
    2.1.3.1. Linear Difference Equations
    2.1.3.2. System of Linear Difference Equations
    2.1.3.3. Non-Linear Difference Equations
    2.2. Introduction to Discrete Models
    2.3. Linear Models
    2.3.1. Population Model Involving Growth
    2.3.2. Newton’s Law of Cooling
    2.3.3. Bank Account Problem
    2.3.4. Drug Delivery Problem
    2.3.5. Harrod Model (Economic Model)
    2.3.6. Arms Race Model
    2.3.7. Lanchester’s Combat Model
    2.3.8. Linear Predator–Prey Model
    2.4. Non-Linear Models
    2.4.1. Density-Dependent Growth Models
    2.4.1.1. Logistic Model
    2.4.1.2. Richer’s Model
    2.4.2. The Learning Model
    2.4.3. Dynamics of Alcohol: A Mathematical Model
    2.4.4. Two Species Competition Model
    2.4.5. 2-cycles
    2.4.6. Stability of 2-cycles
    2.4.7. 3-cycles
    2.5. Bifurcations in Discrete Models
    2.6. Chaos in Discrete Models
    2.6.1. Criteria of Chaos for Discrete Dynamical System
    2.6.2. Quantification of Chaos: Lyapunov Exponent
    2.7. Miscellaneous Examples
    2.8. Mathematica Codes
    2.8.1. Lanchester’s Combat Model (Figure 2.12(a))
    2.8.2. Lyapunov Exponent (Figure 2.20(a))
    2.8.3. Two Species Competition Model (Figure 2.16(a))
    2.8.4. Saddle-Node Bifurcation (Figure 2.18(a))
    2.8.5. Neimark-Sacker Bifurcation (Figure 2.19)
    2.9. Matlab Codes
    2.9.1. Lanchester’s Combat Model (Figure 2.12(a))
    2.9.2. Saddle-Node Bifurcation (Figure 2.18(a))
    2.9.3. Chaotic Behavior in Tent Map (Figure 2.20(b))
    2.9.4. 2D Bifurcation (Figure 2.31)
    2.10. Exercises
    2.11. Projects
    3. Continuous Models Using Ordinary Differential Equations
    3.1. Introduction to Continuous Models
    3.2. Steady-State Solution
    3.3. Stability
    3.3.1. Linearization and Local Stability Analysis
    3.3.2. Lyapunov’s Direct Method
    3.3.2.1. Lyapunov’s Condition for Local Stability
    3.3.2.2. Lyapunov’s Condition for Global Stability
    3.4. Phase Plane Diagrams of Linear Systems
    3.5. Continuous Models
    3.5.1. Carbon Dating
    3.5.2. Drug Distribution in the Body
    3.5.3. Growth and Decay of Current in an L-R Circuit
    3.5.4. Rectilinear Motion under Variable Force
    3.5.5. Mechanical Oscillations
    3.5.5.1. Horizontal Oscillations
    3.5.5.2. Vertical Oscillations
    3.5.5.3. Damped and Forced Oscillations
    3.5.6. Dynamics of Rowing
    3.5.7. Arms Race Models
    3.5.8. Epidemic Models
    3.5.9. Combat Models
    3.5.9.1. Conventional Combat Model
    3.5.9.2. Guerrilla Combat Model
    3.5.9.3. Mixed Combat Model
    3.5.9.4. Guerrilla Combat Model (Revisited)
    3.5.10. Mathematical Model of Love Affair
    3.6. Bifurcations
    3.6.1. Bifurcations in One-Dimension
    3.6.1.1. Saddle-Node Bifurcation
    3.6.1.2. Transcritical Bifurcation
    3.6.1.3. Pitchfork Bifurcation
    3.6.2. Bifurcation in Two-Dimensions
    3.6.2.1. Sotomayar’s Theorem
    3.6.2.2. Saddle-Node Bifurcation
    3.6.2.3. Transcritical Bifurcation
    3.6.2.4. Pitchfork Bifurcation
    3.6.2.5. Hopf Bifurcation
    3.7. Estimation of Model Parameters
    3.7.1. Least Squares Method
    3.7.2. Fitting a Suitable Curve to the Given Data
    3.7.3. Parameter Estimation for ODE
    3.8. Chaos in Continuous Models
    3.8.1. Lyapunov Exponents
    3.8.2. Rossler Systems: Equations for Continuous Chaos
    3.9. Miscellaneous Examples
    3.10. Mathematica Codes
    3.10.1. Stable Node (Figure 3.1(a))
    3.10.2. Stable Node (Figure 3.1(b))
    3.10.3. One-Dimensional Bifurcation (Figure 3.43(a))
    3.10.4. Chaotic Behavior (Figure 3.34(c))
    3.11. Matlab Codes
    3.11.1. Stable Node (Figure 3.1(a))
    3.11.2. Stable Node (Figure 3.1(b))
    3.11.3. Saddle Node Bifurcation (Figure 3.22(a))
    3.11.4. Chaotic Behavior (Figure 3.34(c))
    3.12. Exercises
    3.13. Projects
    4. Spatial Models Using Partial Differential Equations
    4.1. Introduction
    4.2. Heat Flow through a Small Thin Rod (One Dimensional)
    4.3. Two-Dimensional Heat-Equation (Diffusion Equation)
    4.4. Steady Heat Flow: Laplace Equation
    4.4.1. Laplace Equation with Dirichlet’s Boundary Condition
    4.4.2. Laplace Equation with Neumann’s Boundary Condition
    4.5. Wave Equation
    4.5.1. Vibrating String
    4.6. Two-Dimensional Wave Equation
    4.7. Fluid Flow through a Porous Medium
    4.8. Traffic Flow
    4.9. Crime Model
    4.10. Reaction-Diffusion Systems
    4.10.1. Population Dynamics with Diffusion (Single Species)
    4.10.2. Population Dynamics with Diffusion (Two Species)
    4.11. Mathematica Codes
    4.11.1. Heat Equation with Dirichlet’s Condition (Figure 4.2)
    4.11.2. Heat Equation with Neumann’s condition (Figure 4.3)
    4.11.3. One-Dimensional Wave Equation (Figure 4.7)
    4.11.4. Two-Dimensional Heat Equation (Figure 4.4)
    4.11.5. Brusselator Equation One-Dimensional (Figure 4.12)
    4.11.6. Brusselator Equation Two-Dimensional (Figure 4.13)
    4.12. Matlab Codes
    4.12.1. Heat Flow (Figure 4.2)
    4.12.2. Brusselator Equation (Figure 4.12)
    4.12.3. Wave Equation (Figure 4.7(a))
    4.13. Miscellaneous Examples
    4.14. Exercises
    4.15. Project
    5. Modeling with Delay Differential Equations
    5.1. Introduction
    5.2. Linear Stability Analysis
    5.2.1. Linear Stability Criteria
    5.3. Different Models with Delay Differential Equations
    5.3.1. Delayed Protein Degradation
    5.3.2. Football Team Performance Model
    5.3.3. Shower Problem
    5.3.4. Breathing Model
    5.3.5. Housefly Model
    5.3.6. Two-Neuron System
    5.4. Immunotherapy with Interleukin-2, a Study Based on Mathematical Modeling [11] (a Research Problem)
    5.4.1. Background of the Problem
    5.4.2. The Model
    5.4.3. Positivity of the Solution
    5.4.4. Linear Stability Analysis with Delay
    5.4.5. Delay Length Estimation to Preserve Stability
    5.4.6. Numerical Results
    5.4.7. Conclusion
    5.5. Miscellaneous Examples
    5.6. Mathematica Codes
    5.6.1. Delayed Protein Degradation (Figure 5.1)
    5.6.2. Two Neuron System (Figure 5.6)
    5.7. Matlab Codes
    5.7.1. Delayed Protein Degradation (Figure 5.1)
    5.7.2. Two Neuron System (Figure 5.6)
    5.8. Exercises
    5.9. Project
    6. Modeling with Stochastic Differential Equations
    6.1. Introduction
    6.1.1. Random Experiment
    6.1.2. Outcome
    6.1.3. Event
    6.1.4. Sample Space
    6.1.5. Event Space
    6.1.6. Axiomatic definition of Probability
    6.1.7. Probability Function
    6.1.8. Probability Space
    6.1.9. Random Variable
    6.1.10. Sigma Algebra
    6.1.11. Measure
    6.1.12. Probability Measure
    6.1.13. Mean and Variance
    6.1.14. Independent Random Variables
    6.1.15. Gaussian Distribution (Normal Distribution)
    6.1.16. Characteristic Function
    6.1.17. Characteristic Function of Gaussian Distribution
    6.1.18. Inversion Theorem
    6.1.19. Convergence of Random Variables and Limit Theorems
    6.1.20. Stochastic Process
    6.1.21. Markov Process
    6.1.22. Gaussian (Normal) Process
    6.1.23. Wiener Process (Brownian Motion)
    6.1.24. Gaussian White Noise
    6.1.25. Ito Integral:
    6.1.26. Ito’s Formula in One-Dimension
    6.1.27. Stochastic Differential Equation (SDE)
    6.1.28. Stochastic Stability
    6.1.28.1. Stable in Probability
    6.1.28.2. Almost sure Exponential Stability
    6.1.28.3. Moment Exponential Stability
    6.2. Stochastic Models
    6.2.1. Stochastic Logistic Growth
    6.2.2. RLC Electric Circuit with Randomness
    6.2.3. Heston Model
    6.2.4. Two Species Stochastic Competition Model
    6.3. Research Problem: Cancer Self-Remission and Tumor Stability–a Stochastic Approach [145]
    6.3.1. Background of the Problem
    6.3.2. The Deterministic Model
    6.3.3. Equilibria and Local Stability Analysis
    6.3.4. Biological Implications
    6.3.5. The Stochastic Model
    6.3.6. Stochastic Stability of the Positive Equilibrium
    6.3.7. Numerical Results and Biological Interpretations
    6.4. Mathematica Codes
    6.4.1. Stochastic Logistic Growth (Figure 6.2)
    6.4.2. Stochastic Competition Model (Figure 6.4)
    6.5. Matlab Codes
    6.5.1. Stochastic Logistic Growth (Figure 6.2)
    6.5.2. Stochastic Competition Model (Figure 6.4(c))
    6.6. Exercises
    7. Hints and Solutions
    Bibliography
    Index