  We consider a one-dimensional, weakly asymmetric, boundary driven exclusion
process on the interval $[0,N]\cap Z$ in the super-diffusive time scale $N^2
\epsilon^{-1}_N$, where $1\ll \epsilon^{-1}_N \ll N^{1/4}$. We assume that the
external field and the chemical potentials, which fix the density at the
boundaries, evolve smoothly in the macroscopic time scale. We derive an
equation which describes the evolution of the density up to the order
$\epsilon_N$.
