Heat equation

Objective. Solve and interpret diffusion models.

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2. Worked example
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Learn: the key idea

The heat equation uₜ = αuₓₓ models how temperature or concentration diffuses over time, smoothing out spatial variations, with α the diffusivity controlling the rate. Solving on a finite rod with fixed-zero ends via separation of variables gives solutions uₙ(x,t) = sin(nπx/L)e^(−α(nπ/L)²t), each mode decaying exponentially faster for higher n, so sharp features smooth out quickest. Example: an initial sine profile sin(πx/L) decays as e^(−α(π/L)²t) while preserving its shape, illustrating that diffusion damps amplitude without changing the spatial pattern of a single mode. General initial data is expanded as a Fourier sine series, each term decaying independently.

Worked example

Solve the heat equation uₜ = 4uₓₓ on 0<x<π with u(0,t)=u(π,t)=0 and initial condition u(x,0) = 3sin(2x). Find u(x,t).

1 $Match initial data to a single eigenfunction: sin(2x) corresponds to n=2 on [0,π]$
2 $Eigenvalue for mode n is λₙ = n², so λ₂ = 4$
3 $Time decay factor is e^(-αλₙt) = e^(-4\cdot4\cdott) = e^(-16t)$
4 $Combine with amplitude 3: u(x,t) = 3sin(2x)e^(-16t)$

Answer: u(x,t) = 3sin(2x)e^(−16t)

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Common mistake. Students expect heat-equation solutions to oscillate in time like wave solutions; because the time dependence is e^(−λt), not sinusoidal, modes only decay, never oscillate.

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