Half-line drift reflection kernel ($K_{\alpha }(x,y,t)=H(x-y+\alpha t,t)-e^{\alpha y}H(x+y+\alpha t,t)$)

The homogeneous Dirichlet boundary condition kernel for $u_t=u_{xx}+\alpha u_x$ on $x>0$. Here $H(r,t)=e^{-r^2/(4t)}/\sqrt{4\pi t}$ is the heat kernel. The exponential reflection weight makes $K_{\alpha }(0,y,t)=0$. Gaussian decay makes its initial-data integral convergent even when the exponentially weighted initial data are not integrable separately.