HEAT EQUATION & HEAT KERNEL
∂ₜu = α∇²u · Fundamental solution · Gaussian diffusion
Diffusivity α
0.20
Time step dt
0.40
Initial condition
Delta function (heat kernel)
Step function
Wide Gaussian
Multi-peak
Sine wave
Boundary
Dirichlet (u=0)
Neumann (∂u/∂x=0)
Periodic
t = 0.00
max u = —
σ = —
G(x,t) = e^{-x²/4αt} / √(4παt)
The
heat kernel
G(x,t) is the fundamental solution: it solves the heat equation with delta initial data.
Any initial condition can be solved by convolution: u(x,t) = G * u₀.
Key property: width grows as
σ ~ √(αt)
— the hallmark of diffusion.
Top:
1D temperature profile over time
Bottom:
2D heat diffusion visualization
Run / Pause
Step ×10
Reset