Neumann heat kernel on a half-line
= Neumann heat kernel on a half-line
{c}
For the <heat equation> $u_t=\frac12u_{xx}$ on $x>0$ with the <Neumann boundary condition> $u_x(t,0)=0$, the method of images gives
$$
K_t^N(x,y)=p_t(x-y)+p_t(x+y),
\qquad
p_t(z)=\frac1{\sqrt{2\pi t}}e^{-z^2/(2t)}.
$$
Thus $u(t,x)=\int_0^\infty K_t^N(x,y)f(y)dy$. This is also the transition kernel of <Reflected Brownian motion>.