1. PDEs: General classification
“Elliptic” Typical: LaPlace’s Eq.
steady-state
gravity, electrostatics
“Parabolic” Typical: Heat Eq.
conduction
“Hyperbolic” Typical: Wave Eq.
vibration, propagation

3. PDEs: discretization and computational molecules
Forward difference
Backward difference
Symmetric difference
Second difference
Mixed partials
Discrete Laplacian
Example: Marching algorithm for
heat equation

4. PDEs: Where do they come from?
Sketch of derivations:
Heat (diffusion) equation
Wave equation
solution:
traveling waves
boundary conditions
eigenmodes
characteristics
 Dan Russell's excellent site

5. PDEs (wrapping up; segue to Fourier analysis, DSP, image processing, etc.)
More on numerical solution
von Neumann stability analysis
Consistency
Example of inconsistency
Nonlinear vibration
Fermi-Pasta-Ulam-Tsingou and
the idea of a numerical experiment
Fourier modes, nonlinear PDEs, solitons

7. Upcoming themes Surprise in science Fourier (modal) analysis
The Fourier transform becomes an algorithm
The FFT, O(n log n) vs. n2, changes life
Sound and images become digital