Duhamel's principle for heat conduction

Remarks

The idea is to view the source as a stream of initial data: at each time $s$, the slice $f(\cdot, s) \, ds$ starts a free evolution of the heat equation. Formally, $\partial_t u = v(x, t; t) + \int_0^t \partial_t v \, ds = f(x, t) + \alpha \Delta u$. The rigorous proof moves derivatives onto $f$ by a change of variables, which is why $f$ is assumed smooth with compact support.

Example: let $n = 1$ and $f(x, t) = g(x) \psi(t)$, where $g$ and $\psi$ are smooth with compact support. Then $u(x, t) = \int_0^t \psi(s) \, (\Phi(\cdot, t - s) * g)(x) \, ds$, where $*$ is convolution in $x$. It is the temperature of an initially cold rod heated by a source of shape $g$ and strength $\psi(t)$.

With initial data $u(\cdot, 0) = g$, one adds the solution $\Phi(\cdot, t) * g$ of the homogeneous problem. It is the same principle as Duhamel's principle for linear ordinary differential equations.

Going deeper

For a student

The principle reduces the inhomogeneous heat equation to the homogeneous one, which the heat kernel solves. The solution operator $g \mapsto \Phi(\cdot, t) * g$ plays the role of the propagator $e^{(t - s) A}$ in the ODE version. In heat transfer texts, the same idea also handles boundary temperatures that vary in time. One superposes the responses to step changes of the boundary data, weighted by their rates of change.