Duhamel's principle

For a linear evolution equation with zero initial data, Duhamel's principle integrates the homogeneous propagator against the forcing time. For the forced heat equation,

For the planar heat equation with point injection of strength $p(s)$, zero initial data gives $\Psi (t,\mathbf{r})={\int }_0^{t}p(s)e^{-r^2/[4D(t-s)]}/[4\pi D(t-s)]ds$. If injection is $p_0$ until $t_0$ and zero later, then for $t>t_0$ this equals $p_0[\mathrm{Ei}(-b/(t-t_0))-\mathrm{Ei}(-b/t)]/(4\pi D)$, where $b=r^2/(4D)$. The exponential integral expansion gives $p_0\ln [t/(t-t_0)]/(4\pi D)$ when $b/(t-t_0)\ll 1$.

On the real line, initial data $\delta (x-2\sqrt{D})$ and a source $-A\delta (x+2\sqrt{D})\delta (t-1)$ produceAt $x=0,t=2$, the two terms cancel exactly for $A=\sqrt{e/2}$.