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.

The one-dimensional linear advection–diffusion equation is solved analytically by using the Laplace integral transform. The solute transport as well as the flow field is considered to be unsteady, both of independent patterns. The solute dispersion occurs through an inhomogeneous semi-infinite medium. Hence, velocity is considered to be an increasing function of the space variable, linearly interpolated in a finite domain in which solute dispersion behaviour is studied. Dispersion is considered to be proportional to the square of the spatial linear function. Thus, the coefficients of the advection–diffusion equation are functions of both the independent variables, but the expression for each coefficient is considered in degenerate form. These coefficients are reduced into constant coefficients with the help of a new space variable, introduced in our earlier works, and new time variables. The source of the solute is considered to be a stationary uniform point source of pulse type.

Some analytical solutions of one-dimensional advection–diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green’s function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant’s mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.

In the present study, analytical solutions of the advection dispersion equation (ADE) with spatially dependent concave and convex dispersivity are obtained within the fractal and the Euclidean frameworks by using the extended Fourier series method. The dispersion coefficient is considered to be proportional to the nth power of a non-homogeneous quadratic spatial function, where the index n is considered to vary between 0 and 1.5 so that the spatial dependence of dispersivity remains within the limit to describe the heterogeneity in the fractal framework. Real values like n ¼ 0.5 and 1.5 are considered to delineate heterogeneity of the aquifer in the fractal framework, whereas integral values like n = 1 represent thesame in the Euclidean sense. A concave or convex variation is free from demanding a limiting value as in the case of linear variation, hence it is more appropriate in the ambience of many disciplines in which ADE is used. In this study, concentration at the source site remains uniform until the source is present and becomes zero once it is annihilated forever. The analytical solutions, validated through the respective numerical solutions, are obtained in the form of an extended Fourier series with only first five terms. They are convergent to the desired concentration pattern and are stable with the Peclet number. It has been possible because of the formulation of a new Sturm–Liouville problem with advective information. The analytical solutions obtained in this paper are novel.