  In this paper we study high order expansions of chart maps for local finite
dimensional unstable manifolds of hyperbolic equilibrium solutions of scalar
parabolic partial differential equations. Our approach is based on studying an
infinitesimal invariance equation for the chart map that recovers the dynamics
on the manifold in terms of a simple conjugacy. We develop formal series
solutions for the invariance equation and efficient numerical methods for
computing the series coefficients to any desired finite order. We show, under
mild non-resonance conditions, that the formal series expansion converges in a
small enough neighborhood of the equilibrium. An a-posteriori computer assisted
argument proves convergence in larger neighborhoods. We implement the method
for a spatially inhomogeneous Fisher's equation and numerically compute and
validate high order expansions of some local unstable manifolds for morse index
one and two. We also provide a computer assisted existence proof of a
saddle-to-sink heteroclinic connecting orbit.
