Open string mirror maps from Picard- Fuchs equations on relative cohomology

A method for computing the open string mirror map and superpotential, using an extended set of Picard- Fuchs equations, is presented. This is based on techniques used by Lerche, Mayr and Warner. For X a toric hypersurface and Y a hypersurface in X, the mirror map and superpotential are written down explicitly. As an example, the case of K_{P^2} is worked out and shown to agree with the literature