Mathematical Physics with Partial Differential Equations – James Kirkwood – 1st Edition

Chapter 1. Preliminaries
1-1. Self-Adjoint Operators
1-2. Curvilinear Coordinates
1-3. Approximate Identities and the Dirac-δ Function
1-4. The Issue of Convergence
1-5. Some Important Integration Formulas

Chapter 2. Vector Calculus
2-1. Vector Integration
2-2. Divergence and Curl
2-3. Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem

Chapter 3. Green’s Functions
3-1. Construction of Green’s Function using the Dirac-δ Function
3-2. Construction of Green’s Function using Variation of Parameters
3-3. Construction of Green’s Function from Eigenfunctions
3-4. More General Boundary Conditions
3-5. The Fredholm Alternative (Or, what if 0 is an Eigenvalue?)
3-6. Green’s function for the Laplacian in Higher Dimensions

Chapter 4. Fourier Series
4-1. Basic Definitions
4-2. Methods of Convergence of Fourier Series
4-3. The Exponential Form of Fourier Series
4-4. Fourier Sine and Cosine Series
4-5. Double Fourier Series

Chapter 5. Three Important Equations
5-1. Laplace’s Equation
5-2. Derivation of the Heat Equation in One Dimension
5-3. Derivation of the Wave equation in One Dimension
5-4. An Explicit Solution of the Wave Equation
5-5. Converting Second-Order PDEs to Standard Form

Chapter 6. Sturm-Liouville Theory
6-1. The Self-Adjoint Property of a Sturm-Liouville Equation
6-2. Completeness of Eigenfunctions for Sturm-Liouville Equations
6-3. Uniform Convergence of Fourier Series

Chapter 7. Separation of Variables in Cartesian Coordinates
7-1. Solving Laplace’s Equation on a Rectangle
7-2. Laplace’s Equation on a Cube
7-3. Solving the Wave Equation in One Dimension by Separation of Variables
7-4. Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables
7-5. Solving the Heat Equation in One Dimension using Separation of Variables
7-6. Steady State of the Heat equation
7-7. Checking the Validity of the Solution

Chapter 8. Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables
8-1. The Solution to Bessel’s Equation in Cylindrical Coordinates
8-2. Solving Laplace’s Equation in Cylindrical Coordinates using Separation of Variables
8-3. The Wave Equation on a Disk (Drum Head Problem)
8-4. The Heat Equation on a Disk

Chapter 9. Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables
9-1. An Example Where Legendre Equations Arise
9-2. The Solution to Bessel’s Equation in Spherical Coordinates
9-3. Legendre’s Equation and its Solutions
9-4. Associated Legendre Functions
9-5. Laplace’s Equation in Spherical Coordinates

Chapter 10. The Fourier Transform
10-1. The Fourier Transform as a Decomposition
10-2. The Fourier Transform from the Fourier Series
10-3. Some Properties of the Fourier Transform
10-4. Solving Partial Differential Equations using the Fourier Transform
10-5. The Spectrum of the Negative Laplacian in One Dimension
10-6. The Fourier Transform in Three Dimensions

Chapter 11. The Laplace Transform
11-1. Properties of the Laplace Transform
11-2. Solving Differential Equations using the Laplace Transform
11-3. Solving the Heat Equation using the Laplace Transform
11-4. The Wave Equation and the Laplace Transform

Chapter 12. Solving PDEs with Green’s Functions
12-1. Solving the Heat Equation using Green’s Function
12-2. The Method of Images
12-3. Green’s Function for the Wave Equation
12-4. Green’s Function and Poisson’s Equation