Molecular transverse diffusion through unsteady and heterogeneous medium is accounted for in solute mass transport originating from a uniform pulse-type stationary point-source. The corresponding two-dimensional advection-dispersion equation with variable coefficients is solved by the explicit finite difference method. The heterogeneity of the medium is described by a position dependent linear non-homogeneous expression for velocity with unsteady exponential variation with time. Variation of the dispersion parameter due to heterogeneity is considered proportional to square of the velocity. Results are compared to analytical solutions reported in the literature and good agreement is found. The explicit finite difference method is shown to be effective and accurate for solving the related two-dimensional advection-dispersion equation with variable coefficients in semi-infinite media, which is especially important when arbitrary initial and boundary conditions are required. (c) 2013 Elsevier Ltd. All rights reserved.