0 Preliminaries
0.1 Partial Differential Equations
0.2 Separation of Variables
0.3 Orthogonal Functions
1 Fourier Series
1.1 Definitions and Examples
1.2 Convergence of Fourier Series
1.3 Uniform Convergence and the Gibbs Phenomenon
1.4 Parseval's Theorem and Mean Square Error
1.5 Complex Form of a Fourier Series
1.6 Sturm-Liouville Eigenvalue Problems
2 Boundary-Value Problems in Rectangular Coordinates
2.1 The Heat Equation
2.2 Homogeneous Boundary Conditions
2.3 Non-homogeneous Boundary Conditions
2.4 The Vibrating String
2.5 Applications of Multiple Fourier Series
3 Boundary-Value Problems in Cylinderical Coordinates
3.1 Laplace's Equation and Applications
3.2 Bessel Functions
3.3 The Vibrating Drumhead
3.4 Heat Flow in the Infinite Cylinder
3.5 Heat Flow in the Finite Cylinder
4 Boundary-Value Problems in Spherical Coordinates
4.1 Spherically Symmetric Solutions
4.2 Legendre Functions and Spherical Bessel Functions
4.3 Laplace's Equation in Spherical Coordinates
5 Fourier Transforms and Applications
5.1 Basic Properties of the Fourier Transform
5.2 Solution of the Heat Equation for an Infinite Rod
5.3 Solutions of the Wave Equation and Laplace's Equation
5.4 Solution of the Telegraph Equation
6 Asymptotic Analysis
6.1 Asymptotic Analysis of the Factorial Function
6.2 Integration by Parts
6.3 Laplace's Method
6.4 The Method of Stationary Phase
6.5 Asymptotic Expansions
6.6 Asymptotic Analysis of the Telegraph Equation
7 Numerical Analysis
7.1 Numerical Analysis of Ordinary Differential Equations
7.2 The One-Dimensional Heat Equation
7.3 Equations in Several Dimensions
7.4 Variational Methods
7.5 Approximate Methods of Ritz and Kantorovich
7.6 Orthogonality Methods
8 Green's Functions
8.1 Green's Functions for Ordinary Differential Equations
8.2 The Three-Dimensional Poisson Equation
8.3 Two-Dimensional Problems
8.4 Green's Function for the Heat Equation
8.5 Green's Function for the Wave Equation
Appendixes
Answers to Selected Exercises
Index
About the Author
9 Appendixes
10 Answers to Selected Exercises