Duhamel's principle for the heat equation

So far the heat equation has had no internal heat sources — just an initial profile that spreads. But what if a heater is switched on inside the bar, adding heat continuously over time? Duhamel's principle is the elegant bookkeeping trick that builds the solution with sources out of the solutions without sources, by treating the source as a stream of tiny instantaneous pulses.

Consider the inhomogeneous heat equation u_t = k u_xx + f(x,t), with f the source (heat added per unit time), and zero initial data for cleanliness. Duhamel's idea: the heat injected during a brief interval around time s acts like a fresh initial condition f(x,s) ds switched on at time s, which then diffuses forward on its own according to the source-free equation for the remaining time t - s. By linearity, superpose all these contributions. If S(t) denotes the operator that evolves initial data forward by time t under the homogeneous heat equation (convolution with the heat kernel), then u(x,t) = integral from 0 to t of S(t - s) f(x,s) ds. In plain steps: freeze the source at each instant s, let that frozen pulse diffuse for the leftover time t - s, and add up over all s from 0 to t. It is the PDE version of the variation-of-parameters formula you met for ODEs.

Duhamel's principle is a workhorse: it reduces a forced problem to repeated use of the unforced solver you already understand, and it makes clear that the response to a source is the source convolved in time with the system's impulse response (the heat kernel). It is closely tied to Green's functions — the causal Green's function of the heat equation is exactly the kernel that Duhamel's formula integrates against. Caveat: Duhamel as stated handles the source term with zero initial data; the full solution adds the separately-evolved initial condition on top (by superposition). And it relies on linearity — for a nonlinear source the simple superposition fails and you need other methods.