Elementary Partial Differential Equations I Topics

Biomathematics and Applied and Computational Mathematics at Florida State University

References:

Partial Differential Equations: an introduction, by Walter A. Strauss, John Wiley, 1992
Partial Differential Equations, by L. C. Evans, American Mathematical Society, 1998

Topics:

Solution methods for linear PDEs

Separation of variables, eigenfunction expansion, and Fourier series
D'Alembert's solution for the wave equation
Boundary conditions: Dirichlet, Neumann, mixed (Robin), periodic
Solving inhomogeneous PDEs (inhomogeneity in either the PDE or in the BCs)
Higher order problems, e.g. the Euler-Bernoulli beam equation

Rudimentary functional analysis

Vector spaces, both finite and infinite dimensional (e.g. function spaces)
Inner products, projection, norms (including L² and L^p)
Function space examples: L^p, ℂⁿ, etc.
Linear differential operators, self-adjoint operators

Convergence in function spaces and completeness of eigenfunctions

Point-wise, uniform, and L² convergence
Bessel's inequality and Parseval's identity
The Dirichlet kernel and completeness results
Gibbs phenomenon and (briefly) Fourier decay-rate theorems

PDEs in higher spatial dimensions

Eigenvalues and eigenfunctions of the Laplacian in ℝ^d
Eigenfunction solutions in rectangular domains ⊂ ℝ^d
Polar coordinates and Bessel functions
Spherical coordinates and spherical harmonics (brief)

Properties of the Laplace and Poisson equation

Harmonic functions: maximum principle and mean-value property in ℝ^d
Poisson's formula
Existence and uniqueness results for the Laplace/Poisson equation with Dirichlet/Neumann/mixed boundary conditions.