Heat Equation & Separation of Variables: Full Solution

The heat equation ut = α2uxx describes how temperature u(x, t) spreads along a rod, where α2 is the thermal diffusivity of the material. Separation of variables solves it by assuming a product form u(x, t) = X(x)·T(t); the PDE then splits into two ordinary differential equations, one in space and one in time.

This technique is the workhorse for linear PDEs on finite domains. The same separation steps solve the wave equation for vibrating strings and Laplace's equation for steady states, and the resulting sine-series solutions are why Fourier series were invented in the first place.

Key Concepts

• Separation constant −λ: setting X″/X = T′/(α2T) = −λ (negative for bounded solutions) decouples the PDE.
• Spatial eigenvalue problem: X″ + λX = 0 with the boundary conditions (e.g. X(0) = X(L) = 0 for ends held at zero) gives discrete eigenvalues λn and modes Xn.
• Time factor: T′ + α2λT = 0 gives exponential decay e−α2λnt — heat dies out, fastest for high modes.
• Superposition: the full solution is a sum over all modes, u = Σ BnXn(x)e−α2λnt.

Formulas

• Heat equation: ut = α2 uxx, 0 < x < L, t > 0.
• Separated ODEs: X″ + λX = 0; T′ + α2λT = 0.
• For ends held at zero: λn = (nπ/L)2, Xn = sin(nπx/L).
• Solution: u(x,t) = Σn=1∞ Bn sin(nπx/L)·e−(nπα/L)2t, with Bn = (2/L)∫0L f(x)·sin(nπx/L)dx from u(x,0) = f(x).
• Assumptions: constant α2, linear PDE, homogeneous boundary conditions. Limit: non-uniform materials or nonlinear radiation need other methods.

Worked Example

Solve ut = 4uxx (so α2 = 4) on 0 < x < π with u(0,t) = u(π,t) = 0 and initial profile u(x,0) = sin x + 2sin 3x. Here L = π, so λn = n2 and the decay factors are e−4n2t. The initial data is already a sine series with B1 = 1, B3 = 2, all others zero — no integrals needed. Hence u(x,t) = sin x·e−4t + 2sin 3x·e−36t. Verify: at t = 0 it reduces to sin x + 2sin 3x; at x = 0 or π every sine vanishes; and ut = −4sin x·e−4t − 72sin 3x·e−36t equals 4uxx = 4(−sin x·e−4t − 18sin 3x·e−36t). The n = 3 ripple decays nine times faster than the fundamental — the rod smooths itself out.

Common Misconceptions

• Separation works for any PDE. It needs linearity plus a geometry and boundary conditions compatible with product solutions.
• One product X(x)T(t) is the whole answer. Each product is a single mode; the physical solution is the infinite sum that fits the initial data.
• Temperature can oscillate or go negative. The maximum principle forbids it: interior temperature never exceeds the maximum of the initial and boundary values.

Real-World Applications

• Heat conduction in rods, plates, and engine components — the original application.
• Chemical diffusion: ink spreading in water obeys the same equation with a diffusion coefficient.
• Finance: the Black-Scholes option-pricing PDE transforms exactly into the heat equation.

Separation of variables turns ut = α2uxx into the eigenvalue problem X″ + λX = 0 plus exponential time decay, and superposition of the modes fits any initial profile via Fourier coefficients. High-frequency ripples decay fastest, so heat flow smooths. The identical machinery solves the wave and Laplace equations.