Dirichlet boundary-forcing heat-kernel formula
= Dirichlet boundary-forcing heat-kernel formula
{c}
If $K$ is the zero-<Dirichlet heat kernel on an interval> for $w_t=w_{xx}-cw$, <integration by parts> gives
$$
w(x,t)=\int_0^LK(x,y,t)w_0(y)\,dy+\int_0^t[K_y(x,0,t-s)g(s)-K_y(x,L,t-s)h(s)]\,ds.
$$
The opposite endpoint signs are essential. Nonzero boundary values are limits of the full formula; evaluating each homogeneous sine mode at the boundary prematurely loses the forcing.