The heat equation

Fourier’s series solution, the theta-method family (FTCS, Crank-Nicolson, backward Euler), and the method of lines on a 2D plate. An animation shows the fast Fourier modes dying first.

Fourier’s heat equation: a sine series whose modes decay

Fourier's heat equation: a sine series whose modes decay

Crank-Nicolson: large, stable, second-order heat steps

Crank-Nicolson: large, stable, second-order heat steps

The method of lines: a hot plate as a system of ODEs

The method of lines: a hot plate as a system of ODEs

Animated: the heat equation smoothing a rough initial profile

Animated: the heat equation smoothing a rough initial profile