  Within the formalism of the Gross-Pitaevskii equation, we derive effective
one- and two-dimensional equations for cigar- and pancake-shaped dipolar
Bose-Einstein condensates with arbitrary polarization angle. These are based on
an ansatz for the condensate wavefunction whose width in the tightly-confined
direction/s is treated variationally. The equations constitute a coupled
partial differential for the low-dimensional wavefunction and algebraic
equations for the width parameters. This approach accurately predicts the
ground state densities of cigar-shaped and pancake-shaped dipolar Bose-Einstein
condensates, and gives strong agreement with the three-dimensional results,
even as the trapping is relaxed away from the strict quasi-one- and
quasi-two-dimensional regimes. This approach offers a significant improvement
over the standard one- and two-dimensional reduction.
