  We present effective reduced equations for the study of a binary
Bose-Einstein condensate (BEC), where the confining potentials of the two BEC
components have distinct asymmetry so that the components belong to different
space dimensions as in a recent experiment [G. Lamporesi et al., Phys. Rev.
Lett. 104, 153202 (2010)]. Starting from a binary three-dimensional (3D)
Gross-Pitaevskii equation (GPE) and using a Lagrangian variational approach we
derive a binary effective nonlinear Schroedinger equation with components in
different reduced dimensions, e. g., the first component in one dimension and
the second in two dimensions as appropriate to represent a cigar-shaped BEC
coupled to a disk-shaped BEC. We demonstrate that the effective reduced binary
equation, which depend on the geometry of the system, is quite reliable when
compared with the binary 3D GPE and can be efficiently used to perform
numerical simulation and analytical calculation for the investigation of static
and dynamic properties of a binary BEC in mixed dimensions.
