Neumann heat kernel on a half-line

For the heat equation $u_t=\frac{1}{2}{u}_{xx}$ on $x>0$ with the Neumann boundary condition $u_x(t,0)=0$, the method of images givesThus $u(t,x)={\int }_0^{\infty }{K}_t^N(x,y)f(y)dy$. This is also the transition kernel of Reflected Brownian motion.