Dirichlet boundary-forcing heat-kernel formula

If $K$ is the zero-Dirichlet heat kernel on an interval for $w_t=w_{xx}-cw$, integration by parts givesThe 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.

Convolution of this kernel with Dirichlet boundary data supplies the boundary forcing for $u_t=u_{xx}+\alpha u_x$ on $x>0$. It is $-(2{\partial }_x+\alpha )H(x+\alpha t,t)$. For $\alpha \ge 0$, its total time mass is $e^{-\alpha x}$, tending to one as $x\downarrow 0$, and its mass away from zero time tends to zero. Thus the boundary value is recovered through an approximate identity, not by pointwise substitution $x=0$ in the kernel.