  In this study, an implicit-explicit local differential transform method
(IELDTM) based on Taylor series representations is produced for solving 2D and
3D advection-diffusion equations. The parabolic advection-diffusion equations
are reduced to the nonhomogeneous elliptic system of partial differential
equations with the utilization of the Chebyshev spectral collocation approach
in temporal variable. The IELDTM is constructed over 2D and 3D meshes using
continuity equations of the neighbour representations with either explicit or
implicit forms in related directions. The IELDTM is proven to have excellent
convergence properties by experimentally illustrating both h-refinement and
p-refinement outcomes. A distinctive feature of the IELDTM over existing
numerical techniques is the optimization of the local spatial degrees of
freedom. It has been proven that IELDTM provides more accurate results with far
less degrees of freedom than the finite difference, finite element and spectral
methods.
