Chapter 0: Overview

0.1 Overview

Chapter 1: Intro. to Partial Differential Equations

1.1-2 Intro to BVP and Classification of PDEs
1.3-4 Canonical Forms and Intro to PDE for Physical Systems
1.5-7 Boundary Value Problems and Vibrating Systems
1.8 The Conduction of Heat
1.9 Review Exercises

Chapter 2: Graphical Solutions to PDEs

2.1-2 Method of Characteristics for First Order PDEs & Examples
2.3 Method of Characteristics and the Transformation Equations
2.4 D'Alembert's Solution of the Wave Equation
2.5-6 Two Cases for D'Alembert's Solutions
2.7 D'Alembert's Solution and Semi-Infinite Media
2.8 Notes on Boundary Conditions
2.9 Review Exercises

Chapter 3: Lin. Algebra for Boundary Value Probs

3.1 Chapter and Course Motivation
3.2 Vector and Hilbert Spaces
3.3 Euclidean Vector Spaces and Dirac Notation
3.4 The Dirac Delta Function
3.5 An Initial View of Functions as Vectors
3.6 Hilbert Space of Functions
3.7 The Grahm-Schmidt Orthonormalization Procedure
3.8 Fourier Cosine and Sine Series
3.9 Fourier Cosine and Sine Series
3.10 Closure Relations, Kronecker Delta and Dirac Delta Functions
3.11 Fourier Transforms
3.12 Least Squares Approximation
3.13 Approximation to a Function at a Discontinuity
3.14 Periodic Extensions
3.15 Introduction to Direct Product Spaces
3.16 Review Exercises

Chapter 4: Separation of Variables

4.0 Physical Motivation
4.1 Some Basic Concepts
4.2 Separating Variables in the Parabolic Equation
4.3 Separating Variables in the Hyperbolic Equation
4.4 Non-Homogeneous Boundary Conditions
4.5 Note on Interplay Between BCs and the PDE
4.6 The Interval-Doubling Technique
4.7A Method of Eigenfunction Expansion Part 1
4.7B Method of Eigenfunction Expansion Part 2
4.8 Separating Variables in Elliptic Partial Differential Equations
4.9 Separating Variables for an Elliptic Eq using Cylindrical Coords
4.10 Fourier Series in Two Variables
4.11 Review Exercises

Chapter 5: Linear Operators

5.0-1 Introduction to Linear Transformations
5.2 Matrix Representations
5.3 Basis Vector Expansion of a Linear Operator
5.4 Operators and Matrices in Direct Product Space
5.5 Common Matrix Operations
5.6 Vector Space of Linear Operators
5.7 An Algebra of Operators and Commutators
5.8 Unitary Operators and Similarity Transformations
5.9 Hermitian Operators and the Eigenvector Equation
5.10 Self-Adjoint Form of a Second-Order Differential Operator
5.11 A Relation Between Unitary and Hermitian Operators
5.12 Eigenvectors and Eigenvalues for Hermitian Operators
5.13 Finding Eigenvectors, Eigenvalues and Diagonal Matrices
5.14 Theorems for Hermitian Operators
5.15 Second Order Differential Sturm-Liouville Operator
5.16 Examples for Sturm-Liouville Problems
5.17A Ladder and Creation-Annihilation Operators PART 1
5.17B Ladder and Creation-Annihilation Operators PART 2
5.18 Translation Operators

Chapter 6: Integral Transform Methods for BVPs

6.1 Review of Continuous Basis Sets and the Dirac Delta Function
6.2 Some Theorems and Properties for the Fourier Transform
6.3 Solving Boundary Value Problems Using Fourier Transforms
6.4 Fourier Transform Solution and D'Alembert's Solution
6.5 Introduction to the Fourier Sine and Cosine Transforms
6.6 The Space of Functions with Even and Odd Extensions
6.7 Some Sine and Cosine Transforms
6.8 Boundary Value Problems using Sine and Cosine Integrals
6.9 The Laplace Transform
6.10 Simple Examples for the Lapace Transform

Chapter 7: Special Functions

7.1 Legendre Polynomials
7.2 The Radial Solution to Laplace's Equation
7.3 Solving Boundary Value Problems with Legendre Polynomials
7.4 Spherical Harmonics
7.5 Notes on Spherical Harmonics
7.6 Notes on Clebsch-Gordon Coefficients
7.7 Introduction to Bessel Functions
7.8 Solution of Bessel's Equation
7.9 Complete, Orthonormal Sets of Bessel Functions
7.10A Hermite Polynomials and Schrodinger's Equation PART 1
7.10B Hermite Polynomials PART 2

Chapter 8: Special Topics

8.1-2 Intro and Basic Mathematical Idea of the Green Function

Appendix 1: Review of Vector Analysis

A1.1 Scalar and Vector Fields
A1.2 Introduction to the Gradient
A1.3 The Directional Derivative
A1.4 The Divergence
A1.5A Introduction to Curl
A1.5B Green and Stoke's Theorem
A1.6: Review of Coordinate Systems
A1.7 The Laplacian in Cylindrical and Spherical Coordinates
A1.8 Dyads
A1.9 Formulas for Vector Analysis
A1.10 Review Exercises

Appendix 2: 1st Order Ordinary Differential Eqs

A2.1 Review of First Order Differential Equations
A2.2 Review Exercises

Appendix 3: 2nd Order Ordinary Differential Eqs

A3.1-2 Review and Independent Functions
A3.3 Equations with Constant Coefficients
A3.4 Power Series Solution
A3.5 Review Exercises

Appendix 4: Fourier Series and Discontinuity

A4 Comments on Fourier Series and Points of Discontinuity

Appendix 5: Power Series

A5 Properties and Convergence of Series