Wollkind D.J., Dichone B.J. Comprehensive Applied Mathematical Modeling in the Natural and Engineering Sciences. Theoretical Predictions compared with Data

New York: Springer, 2017. — 597 p.This text demonstrates the process of comprehensive applied mathematical modeling through the introduction of various case studies. The case studies are arranged in increasing order of complexity based on the mathematical methods required to analyze the models. The development of these methods is also included, providing a self-contained presentation. To reinforce and supplement the material introduced, original problem sets are offered involving case studies closely related to the ones presented. With this style, the text’s perspective, scope, and completeness of the subject matter are considered unique.
Having grown out of four self-contained courses taught by the authors, this text will be of use in a two-semester sequence for advanced undergraduate and beginning graduate students, requiring rudimentary knowledge of advanced calculus and differential equations, along with a basic understanding of some simple physical and biological scientific principles.What is Comprehensive Applied Mathematical Modeling?
What is the Rationale for this Book?
Canonical Projectile Problem: Finding the Escape Velocity of the Earth
Newton’s Second Law and the Basic Governing Projectile Equation of Motion
Exact Solution Method Involving Velocity as a Function of Altitude
Selection of Scale Factors and Introduction of Nondimensional Variables
Pastoral Interlude: Regular Perturbation Theory of Ordinary Differential Equations
Approximate Solution to the Projectile Problem Involving Regular Perturbation Theory
Energy Method of Solution
Problems
Of Mites and Models
Temperature-Rate Phenomena in Arthropods
Pastoral Interlude: Singular Perturbation Theory of Ordinary Differential Equations
Closed-Form Temperature-Rate Relationships for Mites Using Singular Perturbation Theory
Composite May Predator–Prey Mite Model
Pastoral Interlude: Linear Stability Analysis of the Community Equilibrium Point for a General Predator–Prey Model Involving Slopes of the Isoclines and Exploitation Parameters
Linear Stability Analysis of the Temperature-Dependent Composite May Predator–Prey Mite Model
Pastoral Interlude: Global Stability Behavior of Kolmogorov-Type Predator–Prey Systems and a Limit Cycle Example
Global Stability Behavior of the Temperature-Dependent Composite May Predator–Prey Mite Model
Problems
Canonical Soap Film Problem
Soap Films and Minimal Surfaces
Pastoral Interlude: Equation Satisfied by Catenaries
Pastoral Interlude: Surface Area of Volumes of Revolution
Pastoral Interlude: Calculus of Variations
Calculus of Variations Application to Minimal Soap Film Surfaces
Pastoral Interlude: Envelope of a One-Parameter Family of Curves
Diagrammatic Results for the Canonical Soap Film Problem
Problem
Heat Conduction in a Finite Bar with a Linear Source
Heat Equation in Nonmoving Continua, Divergence Theorem, and DuBois–Reymond Lemma
Equation of State, Constitutive Relations, and Boundary and Initial Conditions
Separation of Variables Solution
Pastoral Interlude: Fourier Series
Fourier Series Application to Heat Conduction in the Finite Bar
Long-Time Behavior of Solution
Problems
Heat Conduction in a Semi-Infinite Bar
Governing Equation and Boundary and Initial Conditions for Impulsive Heat Conduction
Pastoral Interlude: The Buckingham Pi Theorem and Similarity Solutions of PDE’s
Similarity Solution Method
Pastoral Interlude: Asymptotic Series Revisited
Pastoral Interlude: The Complementary Error Function and Watson’s Lemma
Spatial and Temporal Behavior of the Exact Solution
Approximate Solution Method
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
Problems
Initiation of Cellular Slime Mold Aggregation Viewed as an InstabilityPastoral Interlude: Divergence Theorem Revisited, Stokes Theorem, and Green’s Theorem
Formulation of the Problem
Simplified Model of Aggregation
Linear Stability Analysis of its Uniform State
Pastoral Interlude: Fourier Integrals and Laplace Transforms
Satisfaction of Initial Conditions
Mechanistic Interpretation of the Linear Instability Aggregative Criterion
Problem
Chemical Turing Patterns and Diffusive Instabilities
Brusselator Reaction–Diffusion Activator–Inhibitor Model System
Community Equilibrium Point and its Linear Stability Analysis
Diffusive Instabilities and Chemical Turing Pattern Formation
Extension to the Brusselator/Immobilizer Model System
Nonlinear Stability Theory: An Overview
Problems
Governing Equations of Fluid Mechanics
Continuum Hypothesis, Substantial Derivative, and Reynolds Transport Theorem
Conservation of Mass and Continuity Equation
Balance of Linear and Angular Momentum and Conservation of Energy
Constitutive Relation
Equations of State
Problems
Boundary Conditions for Fluid Mechanics
No-Penetration and No-Slip or Adherence Boundary Conditions
Relative Normal Speed and Kinematic Boundary Condition
Jump Conditions at Surfaces of Discontinuity for Mass, Momentum, and Energy
Problems
Subsonic Sound Waves Viewed as a Linear Perturbation in an Inviscid Fluid
Governing Equations of Motion
Linear Perturbation Analysis of its Homogeneous Static Solution
Pastoral Interlude: Characteristic Coordinates
D’Alembert’s Method of Solution of its Wave Equation Formulation
Physical Interpretation of that Solution
Problems
Potential Flow Past a Circular Cylinder of a Homogeneous Inviscid Fluid
Governing Equations of Motion
Pastoral Interlude: Calculus of Variations Method of Change of Variables
Governing Laplace’s Equation in Circular Polar Coordinates
Separation of Variables Solution
D’Alembert’s Paradox
Pastoral Interlude: Leibniz’s Rule of Differentiation
Pastoral Interlude: Legendre Polynomials
Problems
Viscous Fluid Flows
Navier–Stokes Equations in Cartesian and Cylindrical Coordinates
Plane Couette and Poiseuille Flows
Couette and Poiseuille Flows
Small-Gap Regular Perturbation Expansion of Couette Flow
Problems
Blasius Flow Past a Flat Plate
Pastoral Interlude: Singular Perturbation Theory Revisited
Governing Equations of Motion, Vorticity, and the Stream Function
Free-Stream and Boundary-Layer Solutions
Parameters of the Boundary Layer
Physical Interpretation
Problems
Rayleigh–Bénard Natural Convection Problem
Governing Boussinesq Equations of Motion
Simplified Rayleigh Model
Pure Conduction Solution and its Perturbation System
Normal-Mode Linear Stability Analysis
Satisfaction of Initial Conditions
Rayleigh Stability Criterion and Comparison with Experiment
Problems
Heat Conduction in a Finite Bar with a Nonlinear Source
Nondimensional Governing Equation
Linear Stability Analysis
One-Dimensional Planform Stuart-Watson Method of Nonlinear Stability Theory
Truncated Landau Equation
Pattern Formation Results
Problem
Nonlinear Optical Ring-Cavity Model Driven by a Gas Laser
Maxwell–Bloch Governing Equations
Striped Planform Stuart-Watson Expansion
Hexagonal Planform Stuart-Watson Expansion
Pattern Formation Results
Problems
Vegetative Flat Dryland Rhombic Pattern Formation Driven by Root Suction
Basic Governing Equations and a Simplified Model
Equilibrium Points and their Linear Stability
Striped Planform Stuart-Watson Expansion
Rhombic Planform Stuart-Watson Expansion
Pattern Formation Results, Root Suction Characteristic Curve, and an Aridity Classification Scheme
Problems
Calculus of Variations Revisited Plus the Gamma and Bessel Functions
Pastoral Interlude: Euler–Lagrange Equations for Constrained Optimization
Queen Dido’s Problem
Hamilton’s Principle for Conservative Forces of Particle Mass Systems
Derivation of the One-Dimensional Elastic String Equation
Pastoral Interlude: Gamma Function
Laplace’s Method and Stirling’s Formula
Pastoral Interlude: Bessel Functions
Method of Stationary Phase and Asymptotic Representation of Bessel Functions
An Eigenvalue Problem Involving the Bessel Function of the First Kind of Order Zero
Problems
Alternate Methods of Solution for Heat and Wave Equation Problems
Laplace Transform Method of Solution for Heat Conduction in a Semi-Infinite Bar
Pastoral Interlude: Dirac Delta Function
Laplace Transform Method of Solution for Heat Conduction in an Infinite Bar
Fourier Integral Method of Solution for the Sound Wave Problem
Problems
Finite Mathematical Models
Discrete-Time Population Dynamics: Fibonacci Sequence
Minimum Fraction of Popular Votes Necessary to Elect the American President
Financial Mathematics: Compound Interest, Annuities, and Mortgages
Problems
Concluding Capstone Problems
Viral Dynamics
Self-Gravitational Instabilities
Chemically Driven Convection
Complex Form of Nonlinear Stability Expansions
The Black–Scholes Equation
Age-Structured Discrete-Time American Dipper Population Model