Differential Equations: A Toolbox for modeling the world

Foreword
Preface
1 Why Study Differential Equations?
	1.1 The 2008 Olympic 100-Meter Dash
		1.1.1 Usain Bolt's Olympic Victory
		1.1.2 Modeling a Sprint
		1.1.3 The Hill-Keller Differential Equation
	1.2 Intracochlear Drug Delivery
		1.2.1 The Challenge of Hearing Loss
		1.2.2 A Compartmental Model for the Cochlea
		1.2.3 The Differential Equation
	1.3 Population Growth and Fishery Management
		1.3.1 The Need to Manage Fish Harvesting
		1.3.2 Modeling Fish Population
		1.3.3 Modeling Harvesting
		1.3.4 Parameter Estimation and Harvesting
	1.4 Where Do We Go from Here?
		1.4.1 A Toolbox for Describing the World
		1.4.2 Some Terminology
		1.4.3 You Already Know How to Solve Some Differential Equations
		1.4.4 Exercises
	1.5 The Blessing of Dimensionality
		1.5.1 Definition of Dimension
		1.5.2 The Algebra of Dimension
		1.5.3 Derivatives, Integrals, Elementary Functions
		1.5.4 Unit-Free Equations and Bending the Rules
		1.5.5 Using Dimension to Find Plausible Models
		1.5.6 Other Dimensions
		1.5.7 Exercises
	1.6 Modeling Projects
		1.6.1 Project: Hang Time
		1.6.2 Project: Money Matters
		1.6.3 Project: Ant Tunneling
2 First-Order Equations
	2.1 First-Order Linear Equations
		2.1.1 Example: Solving the Hill-Keller Equation as a Linear ODE
		2.1.2 A General Procedure for Solving Linear ODEs
		2.1.3 Some Common First-Order Linear Models
		2.1.4 Exercises
	2.2 Separable Equations
		2.2.1 Application: Falling Objects
		2.2.2 Separation of Variables: A First Example
		2.2.3 The General Procedure for Separation of Variables
		2.2.4 Example: Solving the Falling Object ODE
		2.2.5 Example: Solving the Logistic Equation
		2.2.6 Exercises
	2.3 Qualitative and Graphical Insights
		2.3.1 Direction Fields
		2.3.2 Autonomous Equations
		2.3.3 Phase Portraits
		2.3.4 Fixed Points and Stability
		2.3.5 Determining the Stability of Fixed Points
		2.3.6 Bifurcations
		2.3.7 Exercises
	2.4 The Existence and Uniqueness of Solutions
		2.4.1 Some Inspiration from Calculus 1
		2.4.2 What Are Solutions to ODEs?
		2.4.3 The Existence-Uniqueness Theorem for ODEs
		2.4.4 Exercises
	2.5 Modeling Projects
		2.5.1 Project: Money Matters 2
		2.5.2 Project: Chemical Kinetics
		2.5.3 Project: A Shot in the Water
3 Numerical Methods for ODEs
	3.1 The Need for Numerics
		3.1.1 Logistic Example: Time-Varying Parameters
		3.1.2 Euler's Method
		3.1.3 Evaluate, Extrapolate, Repeat as Necessary
		3.1.4 The Accuracy of Euler's Method
		3.1.5 Exercises
	3.2 Improvements to Euler's Method
		3.2.1 Improving Euler's Method
		3.2.2 The Improved Euler Method
		3.2.3 Exercises
	3.3 Modern Numerical Methods
		3.3.1 The RK4 Algorithm
		3.3.2 Adaptive Step Sizing and Error Control
		3.3.3 Exercises
	3.4 Parameter Estimation
		3.4.1 Hill-Keller Revisited
		3.4.2 Least-Squares Estimation
		3.4.3 Hill-Keller Again
		3.4.4 Least Squares For ODE Parameter Estimation
		3.4.5 A Cautionary Example
		3.4.6 Exercises
	3.5 Modeling Projects
		3.5.1 Project: Sublimation of Carbon Dioxide
		3.5.2 Project: Fish Harvesting Revisited
		3.5.3 Project: The Mathematics of Marriage
		3.5.4 Project: Shuttlecocks and the Akaike Information Criterion
4 Second-Order Equations
	4.1 Vibration and the Harmonic Oscillator
		4.1.1 The 2010 Chilean Earthquake
		4.1.2 The Harmonic Oscillator
		4.1.3 Initial Conditions
		4.1.4 More Applications of Spring-Mass Models
		4.1.5 Exercises
	4.2 The Harmonic Oscillator
		4.2.1 Solving the Harmonic Oscillator ODE: Examples
		4.2.2 Solving Second-Order Linear ODEs: The General Case
		4.2.3 The Underdamped and Undamped Cases
		4.2.4 The General Underdamped Case
		4.2.5 The Critically Damped Case
		4.2.6 The Existence and Uniqueness of Solutions
		4.2.7 Summary and a Physical Perspective
		4.2.8 Exercises
	4.3 The Forced Harmonic Oscillator
		4.3.1 Solving the Forced Harmonic Oscillator Equation
		4.3.2 Finding a Particular Solution: Undetermined Coefficients
		4.3.3 When the Guess Fails
		4.3.4 Exercises
	4.4 Resonance
		4.4.1 An Example of Resonance
		4.4.2 Periodic Forcing
		4.4.3 Exercises
	4.5 Scaling and Nondimensionalization for ODEs
		4.5.1 Motivation: Nonlinear Springs
		4.5.2 Characteristic Variable Scales
		4.5.3 Nondimensionalization: Logistic Equation Example
		4.5.4 Nondimensionalization: Harvested Logistic Equation Example
		4.5.5 The General Outline for Nondimensional Rescaling
		4.5.6 Back to the Hard Spring
		4.5.7 Exercises
	4.6 Modeling Projects
		4.6.1 Project: Earthquake Modeling
		4.6.2 Project: Stay Tuned—RLC Circuits and Radios
		4.6.3 Project: Parameter Estimation with Second-Order ODEs
		4.6.4 Project: Bike Shock Absorber
		4.6.5 Project: The Pendulum
		4.6.6 Project: The Pendulum 2
5 The Laplace Transform
	5.1 Discontinuous Forcing Functions
		5.1.1 Motivation: Pharmacokinetics
		5.1.2 Complication: Discontinuous Forcing
		5.1.3 Complication: Impulsive Forcing
		5.1.4 Discontinuous Forcing and Transform Methods
		5.1.5 Exercises
	5.2 The Laplace Transform
		5.2.1 Definition of the Laplace Transform
		5.2.2 What Kinds of Functions Can Be Transformed?
		5.2.3 Laplace Transforms of Elementary Functions
		5.2.4 Solving Differential Equations Using Laplace Transforms
		5.2.5 The First Shifting Theorem
		5.2.6 The Inverse Laplace Transform
		5.2.7 The Initial and Final Value Theorems
		5.2.8 Section Summary and Remarks
		5.2.9 Exercises
	5.3 Nonhomogeneous Problems and Discontinuous Forcing Functions
		5.3.1 Some Nonhomogeneous Examples
		5.3.2 Discontinuous Forcing
		5.3.3 The Second Shifting Theorem
		5.3.4 Some More Models and Examples
		5.3.5 Summary and Remarks
		5.3.6 Exercises
	5.4 The Dirac Delta Function
		5.4.1 Motivational Examples
		5.4.2 Definition of the Dirac Delta Function
		5.4.3 Three Models: Money, Masses, and Medication
		5.4.4 The Laplace Transform of the Dirac Delta Function
		5.4.5 Solving ODEs with Dirac Delta Functions
		5.4.6 Summary and a Few Remarks
		5.4.7 Laplace Transform Table
		5.4.8 Exercises
	5.5 Input-Output, Transfer Functions, and Convolution
		5.5.1 A System Identification Problem
		5.5.2 Input-Output Systems
		5.5.3 Convolution
		5.5.4 The Impulse Response and Convolution
		5.5.5 System Identification with Impulsive Input
		5.5.6 Exercises
	5.6 A Taste of Control Theory
		5.6.1 The Need for Control
		5.6.2 Modeling an Incubator
		5.6.3 Open-Loop Control
		5.6.4 Closed-Loop Control
		5.6.5 Proportional-Integral Control
		5.6.6 Proportional-Integral-Derivative Control
		5.6.7 Disturbances
		5.6.8 Summary and Comments
		5.6.9 Exercises
	5.7 Modeling Projects
		5.7.1 Project: Drug Dosage
		5.7.2 Project: Machine Replacement
		5.7.3 Project: Vibration Isolation Table Shakedown
		5.7.4 Project: Segway Scooters and The Inverted Pendulum
6 Linear Systems of Differential Equations
	6.1 Systems of Differential Equations
		6.1.1 Motivation: More Pharmacokinetics
		6.1.2 Existence and Uniqueness
		6.1.3 Exercises
	6.2 Linear Constant-Coefficient Homogeneous Systems of ODEs
		6.2.1 Matrix-Vector Formulation
		6.2.2 Solving the Homogeneous Case
		6.2.3 Complex Eigenvalues
		6.2.4 Defective Matrices
		6.2.5 Exercises
	6.3 Linear Constant-Coefficient Nonhomogeneous Systems of ODEs
		6.3.1 Solving Linear Systems of ODEs with Laplace Transforms
		6.3.2 Undetermined Coefficients for Systems of ODEs
		6.3.3 The Significance of Eigenvalues
		6.3.4 Exercises
	6.4 The Matrix Exponential
		6.4.1 Inspiration
		6.4.2 Definition of the Matrix Exponential
		6.4.3 Properties of the Matrix Exponential
		6.4.4 Solving ODEs with the Matrix Exponential
		6.4.5 Computing The Matrix Exponential: The Diagonal Case
		6.4.6 Computing The Matrix Exponential: The Diagonalizable Case
		6.4.7 Computing The Matrix Exponential: Putzer's Algorithm
		6.4.8 Final Remarks
		6.4.9 Exercises
	6.5 Modeling Projects
		6.5.1 Project: LSD Compartment Model
		6.5.2 Project: Homelessness
		6.5.3 Project: Tuned Mass Dampers
7 Nonlinear Systems of Differential Equations
	7.1 Autonomous Nonlinear Systems and Direction Fields
		7.1.1 Some Nonlinear ODE Models
		7.1.2 Direction Fields
		7.1.3 A Nonlinear Direction Field Example
		7.1.4 Direction Fields in Higher Dimensions
		7.1.5 Exercises
	7.2 Direction Fields and Phase Portraits for Linear Systems
		7.2.1 Direction Fields for Homogeneous Linear Systems
		7.2.2 Application to the LSD Model
		7.2.3 The Equation =Ax+b
		7.2.4 Direction Fields for Larger Systems of ODEs
		7.2.5 Exercises
	7.3 Autonomous Nonlinear Systems and Phase Portraits
		7.3.1 Sketching Phase Portraits for Nonlinear Systems
		7.3.2 Linearizing ODEs at Equilibrium Points
		7.3.3 Exercises
	7.4 Analyzing Systems with Unspecified Parameters
		7.4.1 Sketching Phase Portraits with Unspecified Parameters
		7.4.2 Linearizing the Competing Species Model with General Parameters
		7.4.3 Conclusions for Competing Species
		7.4.4 Higher-Dimensional Systems
		7.4.5 Exercises
	7.5 Numerical Methods for Systems of First Order ODE's
		7.5.1 Extending Basic Numerical Methods to Systems
		7.5.2 Stiff Systems of ODEs
		7.5.3 Implicit Numerical ODE Solvers
		7.5.4 Exercises
	7.6 Additional Techniques for Systems of First Order ODEs
		7.6.1 First Integrals and Conservative Systems
		7.6.2 Lyapunov Functions
		7.6.3 Linearization and the Routh-Hurwitz Theorem
		7.6.4 Exercises
	7.7 Modeling Projects
		7.7.1 Project: Homelessness Revisited
		7.7.2 Project: Predator-Prey Model
		7.7.3 Project: Parameter Estimation for Competing Yeast Species
8 An Introduction to Partial Differential Equations
	8.1 Conservation of Stuff and the Continuity Equation
		8.1.1 Industrial Furnaces and Metal Production
		8.1.2 Conservation of Stuff
		8.1.3 The Continuity Equation
		8.1.4 The Heat Equation
		8.1.5 Some Solutions to the Heat Equation: Separation of Variables and Linearity
		8.1.6 Exercises
	8.2 Fourier Series
		8.2.1 An Example
		8.2.2 Approximating Functions
		8.2.3 The Fourier Cosine Expansion
		8.2.4 The Fourier Sine Expansion
		8.2.5 More on Fourier Series Convergence
		8.2.6 Exercises
	8.3 Solving the Heat Equation
		8.3.1 Homogeneous Dirichlet Conditions
		8.3.2 Insulating Boundary Conditions
		8.3.3 Other Boundary Conditions
		8.3.4 Diffusion
		8.3.5 Solving the Nonhomogeneous Heat or Diffusion Equation
		8.3.6 Exercises
	8.4 The Advection and Wave Equations
		8.4.1 The Advection Equation
		8.4.2 Solution to the Advection Equation
		8.4.3 The Wave Equation
		8.4.4 Solution to the Wave Equation
		8.4.5 The Wave Equation on the Real Line
		8.4.6 Exercises
	8.5 Modeling Projects
		8.5.1 Project: It's a Blast (Furnace)!
		8.5.2 Project: Finding Polluters
		8.5.3 Project: Strung Out
		8.5.4 Project: Frequency Analysis of Signals
		8.5.5 Project: It's All Relative
Appendix A Complex Numbers
	A.1 Motivation and Definition
	A.2 Arithmetic with Complex Numbers
	A.3 Exponentiation of Complex Numbers
	A.4 The Fundamental Theorem of Algebra
	A.5 Partial Fraction Decompositions over the Complex Numbers
	A.6 Additional Exercises
Appendix B Matrix Algebra
	B.1 Linear System of Equations
	B.2 Matrix Algebra
	B.3 Eigenvalues and Eigenvectors
	B.4 The Eigenvalues for a General Two by Two Matrix
	B.5 Diagonalization
	B.6 Additional Exercises
Appendix C Circuits
	C.1 Current, Voltage, and Resistance
	C.2 Capacitors
	C.3 Inductors
	C.4 RLC Circuits
	C.5 Complex-Valued Solutions and Periodic Forcing
	C.6 Impedance in Electrical Circuits
Bibliography
Index
Back Cover