One-dimensional advection-diffusion equation with variable coefficients in semi-infinite media is solved numerically by the explicit finite difference method for two dispersion problems: (i) temporally dependent dispersion along a uniform flow and (ii) spatially dependent dispersion along a non-uniform flow. A uniform pulse type input condition and the initial solute concentration that decreases with distance are considered. Results are compared to analytical solutions reported in the literature and good agreement was found. We have shown that explicit finite difference method is effective and accurate for solving the advection-diffusion 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.