= Neumann heat kernel on an interval
{c}
{title2=$K_N(x,y,t)$}
For $L>0$ and $t>0$, the <Sturm-Liouville eigenfunction expansion> for $u_t=u_{xx}$ on $(0,L)$ with homogeneous <Neumann boundary conditions> gives
$$
K_N(x,y,t)=\frac1L+\frac2L\sum_{n=1}^\infty e^{-(n\pi/L)^2t}\cos\frac{n\pi x}{L}\cos\frac{n\pi y}{L}.
$$
The constant <eigenfunction> mode preserves the spatial <integral>. This is the $\beta=0$ limit of the <weighted Neumann heat kernel with constant drift>, and its boundary inputs have the signs in the <Neumann boundary-forcing formula>.
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