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The role of thermal expansion in diffusion edge flames is investigated numerically in the strained mixing layer configuration. We are able to access both the positive as well as the negative edge flame speed regime when density variation, and therefore full hydrodynamic coupling, is present. The computational approach employs a homotopy method to gradually map the solutions from the computationally simpler constant-density flow to the more challenging variable-density case. Particular attention is paid to the role of boundary conditions and how they, in turn, can induce an undesirable streamwise pressure gradient in the trailing diffusion flame that affects the edge flame speed. A new approach is designed to eliminate this adverse pressure gradient. Previous studies observe that the ratio of the edge flame speed to the premixed stoichiometric laminar flame velocity scales approximately as the square root of the ratio of the cold stream density to the stoichiometric density (which is lower). At least for the small set of parameters investigated here, it is found that the speedup of the normalized edge flame velocity might be superlinear on the density ratio, a fact we attribute to the lack of pressure gradient behind the edge flame. This result is new and complements previous results, for different boundary conditions, which strongly suggest that the edge flame speed is a strong function of the particular hydrodynamic boundary conditions employed in the simulations.