= Resolvent kernel for Neumann advection-diffusion on an interval
{title2=$\mathcal R_p(x,y)$}
For $L>0$, constant $\beta>0$ and $\operatorname{Re}p>0$, on $(0,L)$ put $b=\beta/2$, $\kappa=\sqrt{p+b^2}$ with the <principal square root>, and
$$
u_p(x)=e^{-bx}\left[\cosh(\kappa x)+\frac b\kappa\sinh(\kappa x)\right],\quad
v_p(x)=e^{-b(x-L)}\left[\cosh(\kappa(L-x))-\frac b\kappa\sinh(\kappa(L-x))\right].
$$
These satisfy $(p-\partial_x^2-\beta\partial_x)u_p=(p-\partial_x^2-\beta\partial_x)v_p=0$ and the left and right homogeneous <Neumann boundary conditions>, respectively. The weighted <Wronskian> gives
$$
D_p=e^{\beta x}(u_p'v_p-u_pv_p')=e^{bL}\frac p\kappa\sinh(\kappa L),\qquad
\mathcal R_p(x,y)=\frac{u_p(\min(x,y))v_p(\max(x,y))}{D_p}.
$$
The first <derivative> jumps by $-e^{-\beta y}$, so the resolvent integrates against $e^{\beta y}\,dy$. Its time <Laplace transform> interpretation is $\mathcal R_p=\int_0^\infty e^{-pt}K_\beta(t)dt$. The factor $p$ records the zero <eigenvalue> and constant stationary <eigenfunction>.
Back to article page
