Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media

In the present study one-dimensional advection–diffusion equation with variable coefficients is solved for three dispersion problems: (i) solute dispersion along steady flow through an inhomogeneous medium, (ii) temporally dependent solute dispersion along uniform flow through homogeneous medium and (iii) solute dispersion along temporally dependent flow through inhomogeneous medium. Continuous point sources of uniform and increasing nature are considered in an initially solute free semi-infinite medium. Analytical solutions are obtained using Laplace transformation technique. The inhomogeneity of the medium is expressed by spatially dependent flow. Its velocity is defined by a function interpolated linearly in a finite domain in which concentration values are to be evaluated. The dispersion is considered proportional to square of the spatially dependent velocity. The solutions of the third problem may help understand the concentration dispersion pattern along a sinusoidally varying unsteady flow through an inhomogeneous medium. New independent variables are introduced through separate transformations, in terms of which the advection–diffusion equation in each problem is reduced into the one with the constant coefficients. The effects of spatial and temporal dependence on the concentration dispersion are studied with the help of respective parameters and are shown graphically.