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 Lp)
- Function space examples: Lp, 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 ℝd
- Eigenfunction solutions in rectangular domains ⊂ ℝd
- 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 ℝd
- Poisson's formula
- Existence and uniqueness results for the Laplace/Poisson equation with Dirichlet/Neumann/mixed boundary conditions.
