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