Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain

Analytical solutions are obtained for one-dimensional advection-diffusion equation with variable coefficients in a longitudinal finite initially solute free domain, for two dispersion problems. In the first one, temporally dependent solute dispersion along uniform flow in homogeneous domain is studied. In the second problem the velocity is considered spatially dependent due to the inhomogeneity of the domain and the dispersion is considered proportional to the square of the velocity. The velocity is linearly interpolated to represent small increase in it along the finite domain. This analytical solution is compared with the numerical solution in case the dispersion is proportional to the same linearly interpolated velocity. The input condition is considered continuous of uniform and of increasing nature both. The analytical solutions are obtained by using Laplace transformation technique. In that process new independent space and time variables have been introduced. The effects of the dependency of dispersion with time and the inhomogeneity of the domain on the solute transport are studied separately with the help of graphs.