Differential Equations-A Modern Approach with Wavelets

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface for the Instructor
Preface for the Student
1: What Is a Differential Equation?
	1.1 Introductory Remarks
	1.2 A Taste of Ordinary Differential Equations
	1.3 The Nature of Solutions
	1.4 Separable Equations
	1.5 First-Order, Linear Equations
	1.6 Exact Equations
	1.7 Orthogonal Trajectories and Families of Curves
	1.8 Homogeneous Equations
	1.9 Integrating Factors
	1.10 Reduction of Order
		1.10.1 Dependent Variable Missing
		1.10.2 Independent Variable Missing
	1.11 Hanging Chain
		1.11.1 The Hanging Chain
		1.11.2 Pursuit Curves
	1.12 Electrical Circuits
	1.13 The Design of a Dialysis Machine
	Problems for Review and Discovery
2: Second-Order Linear Equations
	2.1 Second-Order Linear Equations with Constant Coefficients
	2.2 The Method of Undetermined Coefficients
	2.3 The Method of Variation of Parameters
	2.4 The Use of a Known Solution to Find Another
	2.5 Vibrations and Oscillations
		2.5.1 Undamped Simple Harmonic Motion
		2.5.2 Damped Vibrations
		2.5.3 Forced Vibrations
		2.5.4 A Few Remarks about Electricity
	2.6 Newton’s Law of Gravitation and Kepler’s Laws
		2.6.1 Kepler’s Second Law
		2.6.2 Kepler’s First Law
		2.6.3 Kepler’s Third Law
	2.7 Higher-Order Coupled Harmonic Oscillators
	Historical Note
	2.8 Bessel Functions and the Vibrating Membrane
	Problems for Review and Discovery
3: Power Series Solutions and Special Functions
	3.1 Introduction and Review of Power Series
		3.1.1 Review of Power Series
	3.2 Series Solutions of First-Order Differential Equations
	3.3 Second-Order Linear Equations: Ordinary Points
	3.4 Regular Singular Points
	3.5 More on Regular Singular Points
	Historical Note
	Historical Note
	3.6 Steady-State Temperature in a Ball
	Problems for Review and Discovery
4: Sturm–Liouville Problems and Boundary Value Problems
	4.1 What Is a Sturm–Liouville Problem?
	4.2 Analyzing a Sturm–Liouville Problem
	4.3 Applications of the Sturm–Liouville Theory
	4.4 Singular Sturm–Liouville
	4.5 Some Ideas from Quantum Mechanics
	Problems for Review and Discovery
5: Numerical Methods
	5.1 Introductory Remarks
	5.2 The Method of Euler
	5.3 The Error Term
	5.4 An Improved Euler Method
	5.5 The Runge–Kutta Method
	5.6 A Constant Perturbation Method for Linear, Second-Order Equations
	Problems for Review and Discovery
6: Fourier Series: Basic Concepts
	6.1 Fourier Coefficients
	6.2 Some Remarks about Convergence
	6.3 Even and Odd Functions: Cosine and Sine Series
	6.4 Fourier Series on Arbitrary Intervals
	6.5 Orthogonal Functions
	Historical Note
	6.6 Introduction to the Fourier Transform
		6.6.1 Convolution and Fourier Inversion
		6.6.2 The Inverse Fourier Transform
	Problems for Review and Discovery
7: Laplace Transforms
	7.1 Introduction
	7.2 Applications to Differential Equations
	7.3 Derivatives and Integrals of Laplace Transforms
	7.4 Convolutions
		7.4.1 Abel’s Mechanics Problem
	7.5 The Unit Step and Impulse Functions
	Historical Note
	7.6 Flow Initiated by an Impulsively Started Flat Plate
	Problems for Review and Discovery
8: Distributions
	8.1 Schwartz Distributions
		8.1.1 The Topology of the Space S
		8.1.2 Algebraic Properties of Distributions
		8.1.3 The Fourier Transform
		8.1.4 Other Spaces of Distributions
		8.1.5 More on the Topology of D and D′
	Problems for Review and Discovery
9: Wavelets
	9.1 Localization in Both Variables
	9.2 Building a Custom Fourier Analysis
	9.3 The Haar Basis
	9.4 Some Illustrative Examples
	9.5 Construction of a Wavelet Basis
		9.5.1 A Combinatorial Construction of the Daubechies Wavelets
		9.5.2 The Daubechies Wavelets from the Point of View of Fourier Analysis
		9.5.3 Wavelets as an Unconditional Basis
		9.5.4 Wavelets and Almost Diagonalizability
	9.6 The Wavelet Transform
	9.7 More on the Wavelet Transform
		9.7.1 Summary
	9.8 Decomposition and Its Obverse
	9.9 Some Applications
	9.10 Cumulative Energy and Entropy
	Problems for Review and Discovery
10: Partial Differential Equations and Boundary Value Problems
	10.1 Introduction and Historical Remarks
	10.2 Eigenvalues, Eigenfunctions, and the Vibrating String
		10.2.1 Boundary Value Problems
		10.2.2 Derivation of the Wave Equation
		10.2.3 Solution of the Wave Equation
	10.3 The Heat Equation
	10.4 The Dirichlet Problem for a Disc
		10.4.1 The Poisson Integral
	Historical Note
	Historical Note
	Problems for Review and Discovery
Table of Notation
Glossary
Bibliography
Index