The Heat Equation — Fourier's Original 1822 Problem
This is literally why Fourier invented Fourier series. He wanted to predict how heat spreads through a metal bar. By decomposing the initial temperature into sines, he turned an impossible PDE into a trivial sum where each mode decays exponentially at its own rate. High-frequency wiggles die first.
Solution to ∂u/∂t = α ∂²u/∂x² on [0, L], u(0)=u(L)=0
Reference: Wikipedia — Heat equation; Fourier, J. (1822). Théorie analytique de la chaleur, op. cit. — the original. Strauss, W. A. (2007). Partial Differential Equations, 2nd ed., Wiley, §2.4 & §4.1.
🧪 Try these experiments in order
- Set time = 0. You see the initial pulse (a rectangular hot spot in the middle of the bar). It's already drawn with its Fourier-series partial sum, so you'll see Gibbs ripples — same as on tab 2.
- Slowly drag time forward. The sharp edges round off first (high modes dying), then the bump shrinks.
- Drop diffusivity α to its lowest value. Same time slider, much slower decay — this is the difference between aluminium and wood.
- Press ▶ animate to watch the full evolution. Notice the bar tends to a constant low value, not zero — this is energy conservation in action.
Temperature profile u(x, t) along the bar (red = hot, blue = cool)
⚠ Watch out for
- Time too large → every mode has effectively decayed; the bar is uniform. Information about the initial shape is permanently lost. This irreversibility is why the heat equation has a preferred direction in time, unlike the wave equation.
- At t = 0 you'll always see Gibbs ringing — that's the Fourier-series representation, not the true initial pulse.
✅ Do
Use this same eigenmode-decay trick for any diffusion problem (concentration spreading, voltage on RC networks, neutron flux).
❌ Don't
Try to "go back in time" — solving the heat equation backwards is famously ill-posed, tiny numerical noise blows up exponentially.
Where this matters in industry
Thermal management in electronics (CPU heatsinks, EV battery packs), metallurgy & heat-treating, building insulation calculations, weather and climate models (atmosphere as a giant diffusive system), drug delivery kinetics, image blurring (a 2D heat equation is exactly a Gaussian blur), financial mathematics (Black-Scholes is a heat equation in disguise).
🎯 Learning checkpoint
Why does the n=10 harmonic decay 100× faster than the n=1 harmonic? Look at the exponent in the formula and explain.