Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media

A linear advection–diffusion equation with variable coefficients in a one-dimensional semi-infinite medium is solved analytically using a Laplace transformation technique, for two dispersion problems: temporally dependent dispersion along a uniform flow and spatially dependent dispersion along a non-uniform flow. Uniform and varying pulse type input conditions are considered. The variable coefficients in the advection–diffusion equation are reduced into constant coefficients with the help of two transformations which introduce new space and time variables, respectively. It is observed that the temporal dependence of increasing nature causes faster solute transport through the medium than that of decreasing nature. Similarly the effect of inhomogeneity of the medium on the solute transport is studied with the help of a function linearly interpolated in a finite space domain.