Separation of Variables for Partial Differential Equations: An Eigenfunction Approach

 THE POTENTIAL, HEAT AND WAVE EQUATION
Overview
Classification of Second Order Equations
Laplace's and Poisson's Equation
The Heat Equation
The Wave Equation

BASIC APPROXIMATION THEORY
Norms and Inner Products
Projection and Best Approximation
Important Function Spaces

STURM-LIOUVILLE PROBLEMS
Sturm-Liouville Problems for f'' = mf
Sturm-Liouville Problems for Lf = mf
A Sturm-Liouville Problem with an Interface

FOURIER SERIES
Introduction
Convergence
Convergence of Fourier series
Cosine and Sine Series
Operations on Fourier Series
Partial Sums of the Fourier Series and Gibbs Phenomenon

EIGENFUNCTION EXPANSIONS FOR EQUATIONS IN R2

THE ONE DIMENSIONAL DIFFUSION EQUATION
Applications of the Eigenfunction Expansion Method
Convergence of uN(x; t) to the Analytic Solution
Influence of the Boundary Conditions and Duhamel's Solution

THE ONE-DIMENSIONAL WAVE EQUATION
Applications of the Eigenfunction Expansion Method
Convergence of uN(x; t) to the Analytic Solution
Eigenfunction Expansions and Duhamel's Principle

POTENTIAL PROBLEMS IN THE PLANE
Applications of the Eigenfunction Expansion Method
Eigenvalue Problem for the Laplacian in R
Convergence of uN(x; y)

MULTIDIMENSIONAL PROBLEMS
Applications of the Eigenfunction Expansion Method
The Eigenvalue Problem for the Laplacian in R3
Bibliography