The Heat Equation — Mathematics Knowledge System

Overview

The heat equation, uₜ = αu_xx, is the canonical parabolic PDE and models diffusion: how temperature, concentration, or any diffusing quantity spreads out and smooths over time. Unlike the wave equation, disturbances in the heat equation do not travel at a finite speed or preserve their shape — instead sharp features immediately blur, and (in an insulated finite domain) the solution decays toward a spatially uniform steady state. The standard solution technique on a finite rod with fixed-temperature ends is separation of variables: assume u(x,t)=X(x)T(t), which splits the PDE into two ODEs whose product solutions, weighted by a Fourier sine series matched to the initial condition, give the full solution. The exponential decay rate of each Fourier mode is proportional to the square of its mode number, so high-frequency (sharp) features die out fastest — exactly the smoothing behavior diffusion is known for.

Intuition

Drop a spoonful of dye into still water: the sharp, concentrated blob doesn't travel intact like a wave — it immediately starts blurring, spreading its 'sharpness' into the surrounding fluid, and given enough time it becomes a faint, uniform tint everywhere. The heat equation captures exactly this: u_xx measures how much a point's value differs from its neighbors' average, and uₜ=αu_xx says the value moves toward that neighborhood average at a rate proportional to how far it currently deviates. Decomposing the initial temperature profile into Fourier sine modes reveals that each mode decays exponentially, and the wigglier (higher-frequency) modes decay much faster than the smoother ones — which is exactly why diffusion erases fine detail first while broad trends persist longest.

Formal Definition

Definition

The one-dimensional heat equation on a rod of length L with fixed-zero end temperatures, solved by separation of variables into a Fourier sine series:

u_t = \alpha u_{xx}, \qquad u(0,t)=u(L,t)=0

Heat equation, Dirichlet (fixed-temperature) ends

u(x,t) = \sum_{n=1}^{\infty} b_n \sin\!\left(\frac{n\pi x}{L}\right) e^{-\alpha (n\pi/L)^2 t}

General solution (Fourier sine series)

b_n = \frac{2}{L}\int_0^L f(x)\sin\!\left(\frac{n\pi x}{L}\right)dx

Fourier coefficients from initial condition u(x,0)=f(x)

Worked Examples

The initial condition is already a single Fourier sine mode (n=2, L=π), so no integral for the coefficients is needed.
$f(x) = 5\sin(2x) \implies b_2 = 5,\ b_n = 0 \ (n\ne 2)$
Apply the general solution formula with α=1, L=π, n=2.
$u(x,t) = 5\sin(2x)e^{-(2)^2 t} = 5\sin(2x)e^{-4t}$

Answer: u(x,t) = 5 sin(2x)e^(-4t) (verify: u_t = -20sin(2x)e^{-4t}; u_xx = -20sin(2x)e^{-4t}; u_t = u_xx ✓; u(x,0)=5sin(2x) ✓; u(0,t)=0, u(π,t)=5sin(2π)e^{-4t}=0 ✓).

Practice Problems

Difficulty 8/10

Solve u_t = 2u_xx on [0,π] with fixed-zero ends and u(x,0) = 4sin(3x).

Difficulty 7/10

For u_t=u_xx on [0,π] with fixed ends, which decays faster over time: the n=1 mode or the n=5 mode, and why?

Difficulty 8/10

A metal rod of length L=π is insulated along its sides with both ends held at 0°C. Its initial temperature profile is u(x,0)=10sin(x) (a single hump peaking at the midpoint). With thermal diffusivity α=0.5, how long until the peak temperature drops to about 10/e ≈ 3.68°C?

Quiz

Summary

The heat equation u_t = αu_xx is parabolic and models diffusion: solutions smooth out and decay toward a steady state rather than propagating.
Separation of variables on a finite rod with fixed-zero ends gives a Fourier sine series solution with each mode decaying as e^{-α(nπ/L)²t}.
Higher-frequency (larger n) modes decay much faster than low-frequency modes, which is why diffusion erases fine spatial detail first.

References

WebsiteWikipedia — Heat equation