Heat, waves and anything that spreads across space.
- Heat equation
- Wave equation
- Laplace's equation
- Separation of variables
- Fourier series
Syllabus
Unit 1Foundations
What partial differential equations are and how they're classified, first-order equations and characteristics, the Fourier series toolkit, and where the heat and wave equations come from.
Unit 2Separation of Variables
Product solutions and eigenvalue problems, the heat equation with fixed, insulated, periodic and nonzero boundaries, standing waves on a string, and d'Alembert's traveling waves.
Unit 3Laplace's Equation
Steady states and harmonic functions: Laplace's equation on rectangles and disks, the mean value property and the maximum principle.
Unit 4Beyond the Basics
Sturm–Liouville theory, heat in two dimensions, drums and Bessel functions, the Fourier transform on infinite domains, numerical methods, image processing and option pricing.
PDE in Physics
All Physics labs
Vibrating Drumhead
The 2D wave equation
PDEPhysics3D
Heat Diffusion
The heat equation
PDEPhysics3D
A Hot Spot Spreading Around a Brake Disc
Periodic boundary conditions on a ring
PDEPhysics
Where a Piano Hammer Should Strike
The vibrating string: standing waves and harmonics
PDEPhysics
Why Quantum Dots Glow Different Colors
Sturm–Liouville eigenvalue problems: eigenfunctions and orthogonality
PDEPhysics
