Comprehensive applied mathematical modeling in the natural and engineering sciences : theoretical predictions compared with data

목차 (Table of Contents)

• CONTENTS
• 1 Introduction = 1
• 1.1 What is Comprehensive Applied Mathematical Modeling? = 1
• 1.2 What is the Rationale for this Book? = 4
• Part Ⅰ Chapters 2–8
• 2 Canonical Projectile Problem : Finding the Escape Velocity of the Earth = 9
• 2.1 Newton's Second Law and the Basic Governing Projectile Equation of Motion = 9
• 2.2 Exact Solution Method Involving Velocity as a Function of Altitude = 13
• 2.3 Selection of Scale Factors and Introduction of Nondimensional Variables = 15
• 2.4 Pastoral Interlude : Regular Perturbation Theory of Ordinary Differential Equations = 18
• 2.5 Approximate Solution to the Projectile Problem Involving Regular Perturbation Theory = 20
• 2.6 Energy Method of Solution = 24
• Problems = 27
• 3 Of Mites and Models = 31
• 3.1 Temperature-Rate Phenomena in Arthropods = 31
• 3.2 Pastoral Interlude : Singular Perturbation Theory of Ordinary Differential Equations = 34
• 3.3 Closed-Form Temperature-Rate Relationships for Mites Using Singular Perturbation Theory = 45
• 3.4 Composite May Predator–Prey Mite Model = 49
• 3.5 Pastoral Interlude : Linear Stability Analysis of the Community Equilibrium Point for a General Predator–Prey Model Involving Slopes of the Isoclines and Exploitation Parameters = 53
• 3.6 Linear Stability Analysis of the Temperature-Dependent Composite May Predator–Prey Mite Model = 60
• 3.7 Pastoral Interlude : Global Stability Behavior of Kolmogorov-Type Predator–Prey Systems and a Limit Cycle Example = 62
• 3.8 Global Stability Behavior of the Temperature-Dependent Composite May Predator–Prey Mite Model = 67
• Problems = 76
• 4 Canonical Soap Film Problem = 81
• 4.1 Soap Films and Minimal Surfaces = 81
• 4.2 Pastoral Interlude : Equation Satisfied by Catenaries = 82
• 4.3 Pastoral Interlude : Surface Area of Volumes of Revolution = 86
• 4.4 Pastoral Interlude : Calculus of Variations = 89
• 4.5 Calculus of Variations Application to Minimal Soap Film Surfaces = 90
• 4.6 Pastoral Interlude : Envelope of a One-Parameter Family of Curves = 91
• 4.7 Diagrammatic Results for the Canonical Soap Film Problem = 93
• Problem = 96
• 5 Heat Conduction in a Finite Bar with a Linear Source = 99
• 5.1 Heat Equation in Nonmoving Continua, Divergence Theorem, and DuBois–Reymond Lemma = 99
• 5.2 Equation of State, Constitutive Relations, and Boundary and Initial Conditions = 101
• 5.3 Separation of Variables Solution = 104
• 5.4 Pastoral Interlude : Fourier Series = 106
• 5.5 Fourier Series Application to Heat Conduction in the Finite Bar = 109
• 5.6 Long-Time Behavior of Solution = 110
• Problems = 112
• 6 Heat Conduction in a Semi-Infinite Bar = 115
• 6.1 Governing Equation and Boundary and Initial Conditions for Impulsive Heat Conduction = 115
• 6.2 Pastoral Interlude : The Buckingham Pi Theorem and Similarity Solutions of PDE's = 117
• 6.3 Similarity Solution Method = 118
• 6.4 Pastoral Interlude : Asymptotic Series Revisited = 121
• 6.5 Pastoral Interlude : The Complementary Error Function and Watson's Lemma = 124
• 6.6 Spatial and Temporal Behavior of the Exact Solution = 127
• 6.7 Approximate Solution Method = 129
• 6.8 Heat Conduction in Contact with a Reservoir of Oscillating Temperature or Why When it is Summer on the Earth's Surface it is Winter 4.44 meters Under Ground in its Crust = 132
• Problems = 137
• 7 Initiation of Cellular Slime Mold Aggregation Viewed as an Instability = 145
• 7.1 Introduction = 146
• 7.2 Pastoral Interlude : Divergence Theorem Revisited, Stokes Theorem, and Green's Theorem = 147
• 7.3 Formulation of the Problem = 152
• 7.4 Simplified Model of Aggregation = 156
• 7.5 Linear Stability Analysis of its Uniform State = 157
• 7.6 Pastoral Interlude : Fourier Integrals and Laplace Transforms = 160
• 7.7 Satisfaction of Initial Conditions = 163
• 7.8 Mechanistic Interpretation of the Linear Instability Aggregative Criterion = 164
• Problem = 165
• 8 Chemical Turing Patterns and Diffusive Instabilities = 167
• 8.1 Brusselator Reaction–Diffusion Activator–Inhibitor Model System = 167
• 8.2 Community Equilibrium Point and its Linear Stability Analysis = 168
• 8.3 Diffusive Instabilities and Chemical Turing Pattern Formation = 171
• 8.4 Extension to the Brusselator／Immobilizer Model System = 174
• 8.5 Nonlinear Stability Theory : An Overview = 181
• Problems = 184
• Part Ⅱ Chapters 9–14
• 9 Governing Equations of Fluid Mechanics = 189
• 9.1 Continuum Hypothesis, Substantial Derivative, and Reynolds Transport Theorem = 189
• 9.2 Conservation of Mass and Continuity Equation = 196
• 9.3 Balance of Linear and Angular Momentum and Conservation of Energy = 207
• 9.4 Constitutive Relation = 211
• 9.5 Equations of State = 218
• Problems = 223
• 10 Boundary Conditions for Fluid Mechanics = 231
• 10.1 No-Penetration and No-Slip or Adherence Boundary Conditions = 231
• 10.2 Relative Normal Speed and Kinematic Boundary Condition = 233
• 10.3 Jump Conditions at Surfaces of Discontinuity for Mass, Momentum, and Energy = 235
• Problems = 246
• 11 Subsonic Sound Waves Viewed as a Linear Perturbation in an Inviscid Fluid = 251
• 11.1 Governing Equations of Motion = 251
• 11.2 Linear Perturbation Analysis of its Homogeneous Static Solution = 252
• 11.3 Pastoral Interlude : Characteristic Coordinates = 254
• 11.4 D'Alembert's Method of Solution of its Wave Equation Formulation = 256
• 11.5 Physical Interpretation of that Solution = 257
• Problems = 259
• 12 Potential Flow Past a Circular Cylinder of a Homogeneous Inviscid Fluid = 263
• 12.1 Governing Equations of Motion = 263
• 12.2 Pastoral Interlude : Calculus of Variations Method of Change of Variables = 267
• 12.3 Governing Laplace's Equation in Circular Polar Coordinates = 272
• 12.4 Separation of Variables Solution = 273
• 12.5 D'Alembert's Paradox = 275
• 12.6 Pastoral Interlude : Leibniz's Rule of Differentiation = 279
• 12.7 Pastoral Interlude : Legendre Polynomials = 281
• Problems = 295
• 13 Viscous Fluid Flows = 303
• 13.1 Navier–Stokes Equations in Cartesian and Cylindrical Coordinates = 303
• 13.2 Plane Couette and Poiseuille Flows = 306
• 13.3 Couette and Poiseuille Flows = 311
• 13.4 Small-Gap Regular Perturbation Expansion of Couette Flow = 317
• Problems = 325
• 14 Blasius Flow Past a Flat Plate = 329
• 14.1 Pastoral Interlude : Singular Perturbation Theory Revisited = 329
• 14.2 Governing Equations of Motion, Vorticity, and the Stream Function = 339
• 14.3 Free-Stream and Boundary-Layer Solutions = 342
• 14.4 Parameters of the Boundary Layer = 347
• 14.5 Physical Interpretation = 350
• Problems = 354
• Part Ⅲ Chapters 15–18
• 15 Rayleigh–Bénard Natural Convection Problem = 361
• 15.1 Governing Boussinesq Equations of Motion = 361
• 15.2 Simplified Rayleigh Model = 365
• 15.3 Pure Conduction Solution and its Perturbation System = 367
• 15.4 Normal-Mode Linear Stability Analysis = 368
• 15.5 Satisfaction of Initial Conditions = 375
• 15.6 Rayleigh Stability Criterion and Comparison with Experiment = 376
• Problems = 391
• 16 Heat Conduction in a Finite Bar with a Nonlinear Source = 399
• 16.1 Nondimensional Governing Equation = 400
• 16.2 Linear Stability Analysis = 400
• 16.3 One-Dimensional Planform Stuart-Watson Method of Nonlinear Stability Theory = 402
• 16.4 Truncated Landau Equation = 410
• 16.5 Pattern Formation Results = 412
• Problem = 418
• 17 Nonlinear Optical Ring-Cavity Model Driven by a Gas Laser = 423
• 17.1 Maxwell–Bloch Governing Equations = 423
• 17.2 Striped Planform Stuart-Watson Expansion = 430
• 17.3 Hexagonal Planform Stuart-Watson Expansion = 434
• 17.4 Pattern Formation Results = 443
• Problems = 451
• 18 Vegetative Flat Dryland Rhombic Pattern Formation Driven by Root Suction = 457
• 18.1 Basic Governing Equations and a Simplified Model = 458
• 18.2 Equilibrium Points and their Linear Stability = 459
• 18.3 Striped Planform Stuart-Watson Expansion = 464
• 18.4 Rhombic Planform Stuart-Watson Expansion = 467
• 18.5 Pattern Formation Results, Root Suction Characteristic Curve, and an Aridity Classification Scheme = 472
• Problems = 483
• Part Ⅳ Chapters 19–22
• 19 Calculus of Variations Revisited Plus the Gamma and Bessel Functions = 491
• 19.1 Pastoral Interlude : Euler–Lagrange Equations for Constrained Optimization = 491
• 19.2 Queen Dido's Problem = 494
• 19.3 Hamilton's Principle for Conservative Forces of Particle Mass Systems = 496
• 19.4 Derivation of the One-Dimensional Elastic String Equation = 500
• 19.5 Pastoral Interlude : Gamma Function = 502
• 19.6 Laplace's Method and Stirling's Formula = 505
• 19.7 Pastoral Interlude : Bessel Functions = 508
• 19.8 Method of Stationary Phase and Asymptotic Representation of Bessel Functions = 516
• 19.9 An Eigenvalue Problem Involving the Bessel Function of the First Kind of Order Zero = 526
• Problems = 529
• 20 Alternate Methods of Solution for Heat and Wave Equation Problems = 543
• 20.1 Laplace Transform Method of Solution for Heat Conduction in a Semi-Infinite Bar = 543
• 20.2 Pastoral Interlude : Dirac Delta Function = 546
• 20.3 Laplace Transform Method of Solution for Heat Conduction in an Infinite Bar = 549
• 20.4 Fourier Integral Method of Solution for the Sound Wave Problem = 552
• Problems = 554
• 21 Finite Mathematical Models = 557
• 21.1 Discrete-Time Population Dynamics : Fibonacci Sequence = 557
• 21.2 Minimum Fraction of Popular Votes Necessary to Elect the American President = 560
• 21.3 Financial Mathematics : Compound Interest, Annuities, and Mortgages = 564
• Problems = 567
• 22 Concluding Capstone Problems = 573
• 22.1 Viral Dynamics = 573
• 22.2 Self-Gravitational Instabilities = 577
• 22.3 Chemically Driven Convection = 582
• 22.4 Complex Form of Nonlinear Stability Expansions = 586
• 22.5 The Black–Scholes Equation = 588
• 22.6 Age-Structured Discrete-Time American Dipper Population Model = 593
• References = 597
• Index = 605