Heat Equation & Fourier Decomposition
∂T/∂t = α ∇²T · diffusion from hot to cold · Draw on 1D bar
PARAMETERS
Diffusivity α
0.10
Fourier modes
8
INITIAL CONDITIONS (1D)
Sine wave
Step function
Gaussian pulse
Two spikes
↺ Reset 1D
2D HEAT MAP
Hot circle
Hot stripe
Random spots
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Speed
5
Fourier's Law:
∂T/∂t = α ∂²T/∂x²
Solution: each mode decays as
T_n(t) = b_n · e^(−α(nπ/L)²t) sin(nπx/L)
High-frequency modes decay
fastest
(smooth out quickly). Low modes persist longest — this is why long-term temperature distributions are smooth.