Heat equation and diffusion

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

What is it?

$u_t = \alpha\Delta u$ : temperature (or concentration, or probability) flows from high to low and smooths out. Solved by finite differences or Fourier series; Gaussian blur of an image is exactly running this equation.

Formulas

$\frac{\partial u}{\partial t} = \alpha\,\Delta u$; $u_i^{n+1} = u_i^n + \frac{\alpha\,\Delta t}{\Delta x^2}\big(u_{i+1}^n - 2u_i^n + u_{i-1}^n\big), \qquad \frac{\alpha\,\Delta t}{\Delta x^2} \le \frac12$
explicit scheme and its stability condition

Why does it matter?

CPU and battery thermal design, materials processing, and — through the same mathematics — image smoothing and diffusion-based generative models.

The mathematics behind it

Numerical differentiation★★★★★fundamental
Finite-difference solvers replace $\partial^2/\partial x^2$ by the $[1, -2, 1]/h^2$ stencil on a grid.
Partial derivatives★★★★★fundamental
The heat equation $u_t = \alpha(u_{xx} + u_{yy})$ is a relation between partial derivatives.
Laplacian★★★★★fundamental
Diffusion is $u_t = \alpha\Delta u$ .
Partial differential equations★★★★★fundamental
The prototype parabolic PDE.
Fourier series★★★★★historical
Fourier invented his series to solve the heat equation; spectral methods still do.

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