Dirichlet heat kernel on an interval
= Dirichlet heat kernel on an interval
{c}
{title2=$H_L(x,y,t)$}
For the <heat equation> with zero <Dirichlet boundary conditions> on $(0,L)$,
$$
H_L(x,y,t)=\frac2L\sum_{n\ge1}e^{-(n\pi/L)^2t}\sin(n\pi x/L)\sin(n\pi y/L).
$$
It is also obtained by the <method of images>, taking the difference between Gaussian sources at $y+2jL$ and $-y+2jL$. The <eigenfunction expansion> is useful at long times; the image expansion is useful at short times.