∂u/∂t = α ∂²u/∂x² — draw any initial temperature profile and watch it diffuse over time. The solution is a superposition of decaying sinusoidal modes, each with e^(−αn²π²t/L²) decay rate.
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Joseph Fourier (1822) solved the heat equation by decomposing the initial condition into sinusoidal modes. Each mode sin(nπx/L) decays independently at rate e^(−αn²π²t/L²). High frequencies decay much faster — this is why sharp features smooth out first. Dirichlet BC fixes temperature at 0 at both ends; Neumann BC insulates them.