Partial differential equations

Level AdvancedDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Equations for fields that depend on space and time, involving partial derivatives: heat diffusion, waves, fluid flow, electromagnetism, quantum mechanics. Solved numerically by discretizing space (finite differences, finite volumes, finite elements, spectral methods).

Formulas

$u_t = \alpha\,\Delta u, \qquad u_{tt} = c^2\Delta u, \qquad \Delta u = 0$
heat, wave, Laplace: parabolic, hyperbolic, elliptic

Where it shows up in computing

Heat equation and diffusion★★★★★fundamentalPhysics and simulation
The prototype parabolic PDE.
Wave equation★★★★★fundamentalPhysics and simulation
The prototype hyperbolic PDE.
Fluid dynamics and CFD★★★★★fundamentalPhysics and simulation
The Navier–Stokes equations are a nonlinear system of PDEs.
Finite element method★★★★★fundamentalPhysics and simulation
FEM is the dominant method to solve PDEs on complex geometries in engineering.
Image processing and computer vision★★★★★advancedSignals, media and vision
Anisotropic (Perona–Malik) diffusion denoises images while preserving edges by solving a nonlinear heat equation.

Where it shows up in AI

Generative models★★★★★advancedAI and machine learning
Diffusion models are tied to the Fokker–Planck PDE of a noising process, reversed by learning the score $\nabla\log p$ .

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

What is it?

Formulas

Where it shows up in computing

Where it shows up in AI

Where is it used?

⚛ Physics and simulation

ℒ AI and machine learning

∿ Signals, media and vision