Introduction to Qualitative Methods for Differential Equations

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Preliminaries
	0.1  Purpose of Book
	0.2  Experiments and Physical Measurements
	0.3  Mathematics and Physical Mathematics
	0.4  Dynamic Consistency and Mathematical Modeling
	0.5  Physical and Mathematical Equations
	0.6  Local Behavior of Functions
	Problems
	Notes and References
Chapter 1 What Is a Solution?
	1.1  Introduction
	1.2  Radioactive Decay
	1.3  x2+y2=1
	1.4  Mickens–Newton Law of Cooling
	1.5  What Are the Solutions to y+y=0?
	1.6  Approximate Solution to the Heat PDE
	1.7  Discussion
	Problems
	Notes and References
Chapter 2 One-Dimensional Systems
	2.1  Introduction
	2.2  Fixed-Points
	2.3  Sign of the Derivative: One Fixed-Point
	2.4  Two Fixed-Points
	2.5  Linear Stability
	2.6  Applications
		2.6.1  Radioactive Decay
		2.6.2  Logistic Equation
		2.6.3  Gompertz Model
		2.6.4  Draining a Tank
		2.6.5  f(x,t) Depends on t
		2.6.6  Spruce Budworm Population Model
	2.7  Discussion
	Problems
	Notes and References
Chapter 3 Two-Dimensional Dynamical Systems
	3.1  Introduction
	3.2  Definitions
		3.2.1  2-Dim Dynamical System
		3.2.2  Fixed-Points
		3.2.3  Nullclines
		3.2.4  First-Integral and Symmetry Transformations
	3.3  General Features of Trajectories
	3.4  Constructing Phase-Plane Diagrams
	3.5  Linear Stability Analysis
	3.6  Local Behavior of Nonlinear Systems
	3.7  Examples
		3.7.1  Harmonic Oscillator
			Comments
		3.7.2  Damped Harmonic Oscillator
		3.7.3  Nonlinear Cubic Oscillator
		3.7.4  Damped Cube-Root Oscillator
		3.7.5  x+(1+x)x=0
		3.7.6  Simple Predator–Prey Model
		3.7.7  van der Pol Equation
		3.7.8  SIR Model for Disease Spread
	3.8  Discussion
	Problems
	Comments and References
Chapter 4 Sturm–Liouville Problems
	4.1  Introduction
		4.1.1  Elimination of First-Derivative Term
		4.1.2  Liouville–Green Transformation
	4.2  The Vibrating String
		4.2.1  Both Ends Fixed
		4.2.2  One Fixed and One Free Ends
		4.2.3  Both Ends Free
		4.2.4  Summary
	4.3  Separation and Comparison Results
		4.3.1  y(x)+f(x)y(x)=0
	4.4  Sturm–Liouville Problems
		4.4.1  Properties of the Eigenvalues and Eigenfunctions
		4.4.2  Orthogonality of Eigenfunctions
		4.4.3  Expansion of Functions
	4.5  Related Issues
		4.5.1  Reduction to Sturm–Liouville Form
		4.5.2  Fourier Series
		4.5.3  Special Functions
		4.5.4  TISE: Sketches of Wavefunctions
	Problems
	Comments and References
Chapter 5 Partial Differential Equations
	5.1  General Comments
	5.2  Symmetry-Derived PDEs
		5.2.1  Heat Conduction PDE
	Comments
		5.2.2  Wave PDE
		5.2.3  Discussion
	Comments
	5.3  Method of Separation of Variables
		5.3.1  Introduction
		5.3.2  Definition of the Method of SOV
		5.3.3  Examples
	5.4  Traveling Waves
		5.4.1  Burgers’ Equation
		5.4.2  Korteweg de Vries Equation
		5.4.3  Fisher’s Equation
		Comments
		5.4.4  Heat PDE
	Problems
	Notes and References
Chapter 6 Introduction to Bifurcations
	6.1  Introduction
	6.2  Definition
		6.2.1  Bifurcation
	6.3  Examples of Elementary Bifurcations
		6.3.1  Saddle-node Bifurcation
		6.3.2  Transcritical Bifurcation
		6.3.3  Supercritical Pitchfork Bifurcation
		6.3.4  Subcritical Pitchfork Bifurcation
	6.4  Examples from Physics
		6.4.1  Lasers
		6.4.2  Statistical Mechanics and Neural Networks
	6.5  Hopf-Bifurcations
		6.5.1  Introduction
		6.5.2  Hopf-Bifurcation Theorem
		6.5.3  Fixed-Points and Closed Integral Curves
		6.5.4  Two Limit-Cycle Oscillators
	6.6  Resumé
	Problems
	Comments and References
Chapter 7 Applications
	7.1  Estimation of y(0) for a Boundary-Value Problem
		7.1.1  Properties of y(z)
		7.1.2  Approximation to y(z)
		7.1.3  Resume
	7.2  Thomas–Fermi Equation (TFE)
		7.2.1  Exact Results
		7.2.2  Approximate Solutions
		7.2.3  Discussion
	7.3  Traveling-Wave Front Behavior for a PDE Having Square-Root Dynamics
		7.3.1  Variable Scaling
		7.3.2  Traveling Wave Solutions
		7.3.3  Traveling Wave Front Behavior
		7.3.4  Case I
		7.3.5  Case II
		7.3.6  Case III
		7.3.7  Approximation to Traveling Wave Solution
	7.4  Comments on Functional Equation Models of Radioactive Decay and Heat Conduction
	7.5  Approximate Solutions to a Modified, Nonlinear Maxwell–Cattane Equation
		7.5.1  Positivity and Equilibrium Solutions
		7.5.2  Space-Independent Solutions
		7.5.3  Traveling Waves
		7.5.4  Resume
	7.6  Nonlinear Oscillations: An Averaging Method
		7.6.1  First Approximation of Krylov and Bogoliubov
		7.6.2  Higher-Order Corrections
		7.6.3  Examples
		7.6.4  Summary
	7.7  Culling in Predator–Prey Systems
		7.7.1  Predator–Prey Models
		7.7.2  General Properties of Predator–Prey Models
		7.7.3  Culling
		7.7.4  Culling the Predator
		7.7.5  Summary
	7.8  A Linear ODE: y=(x−y)/x2
		7.8.1  Qualitative Analysis
		7.8.2  Construction of an Approximate Solution
		7.8.3  Summary
	7.9  Approximating ‘1’ and ‘0’
		7.9.1  Introduction
		7.9.2  Finite Difference Discretization of a Modified Decay ODE
		7.9.3  d2x/dt2+x3=0
		7.9.4  x+x13=0
		7.9.5  Discussion
	Comments and References
	References to the Exponential Functions
Appendix A
	A.1  Algebraic Relations
		A.1.1  Factors and Expansions
		A.1.2  Quadratic Equations
		A.1.3  Cubic Equations
		A.1.4  Expansions of Selected Functions
	A.2  Trigonometric Relations
		A.2.1  Fundamental Properties
		A.2.2  Sum of Angles Relations
		A.2.3  Product and Sum Relations
		A.2.4  Derivatives and Integrals
		A.2.5  Powers of Trigonometric Functions
	A.3  Hyperbolic Functions
		A.3.1  Definitions and Basic Properties
		A.3.2  Derivatives and Integrals
		A.3.3  Other Important Relations
		A.3.4  Relations between Trigonometric and Hyperbolic Functions
	A.4  Important Calculus Relationships
		A.4.1  Differentiation
		A.4.2  Integration by Parts
		A.4.3  Differentiation of a Definite Integral
	A.5  Even and Odd Functions
	A.6  Absolute Value Function
	A.7  Differential Equations
		A.7.1  General Linear, First-Order Ordinary Differential Equation
		A.7.2  Bernoulli Equations
		A.7.3  Riccati Equation
		A.7.4  Linear, Second-Order Differential Equations with Constant Coefficients
		A.7.5  Fourier Series
Bibliography
Index