Digital call from the heat equation

A cash-or-nothing call pays $1 at time $T$ if $S_T > K$. After the usual substitutions its Black-Scholes value is $V = e^{-r\tau}w(z,\tau)$, where $w$ solves $w_\tau = \tfrac12\sigma^2 w_{zz}$, $z = \ln S + (r - \tfrac12\sigma^2)\tau$, and $w(z,0) = 1$ for $z > \ln K$ and $0$ otherwise.

Solve for $w$ and price the digital with $S = K = 100$, $r = 5\%$, $\sigma = 20\%$, $\tau = 1$.