Heat equation ∂u/∂t = α∇²u (Fourier, 1822): describes diffusion of heat, chemical concentration, or probability.
Solved by explicit finite differences: u(t+dt) = u(t) + α·dt/dx²·(u_left+u_right+u_up+u_down−4u).
The solution smooths out instantly (infinite propagation speed — parabolic PDE).
The profile view shows the temperature cross-section at mid-height.
Try: draw a hot line, watch it diffuse; observe how total energy (integral) is conserved.