Symmetric absorbing-interval survival probability ($S(t)=\frac{4}{\pi }{\sum }_{j\ge 0}\frac{(-1{)}^j}{2j+1}{e}^{-(2j+1{)}^2{\pi }^{2}Dt/(4L^2)}$)

For diffusion starting at the centre of $(-L,L)$ with absorbing endpoints, only even-parity spatial eigenfunctions contribute. Integrating their cosine expansion gives the displayed series. The leading late-time term has prefactor $4/\pi$ and decay rate ${\pi }^{2}D/(4L^2)$; its first relative correction is $-\frac{1}{3}{e}^{-2{\pi }^{2}Dt/L^2}$.