Degree Name

Doctor of Philosophy


School of Mathematics and Applied Statistics


Contaminant transport in soils is described by a system of coupled (soil-water) advection-diffusion equations. This system has few symmetries. However, we first consider the two dimensional partial differential equations (P.D.E.s) describing transport of solutes in saturated soils, i.e. solute transport under steady water flow background. The quest for exact analytic solutions for these equations has continued unabated. The problem is difficult when velocity must be given by the modulus of a potential flow velocity field for incompressible fluids. However, the Laplace preserving transformations from Cartesian to streamline coordinates results in a much simpler form of solute transport equation. Classical symmetry analysis of the transformed equations with point water source and point vortex water flow results in a rich array of classical point symmetries. Exploitation of symmetry properties and other transformation techniques lead to a number of exotic exact analytic solutions.

Next, we examine solutions for the one dimensional solute transport during steady evaporation from a water table. If we classify the symmetry-bearing cases of the coefficient functions within a single solute transport equation, then some of the cases are compatible with special solutions for soil-water flow. This may be viewed as reduction of a system of couple soil-water equations by conditional or nonclassical symmetries. In most known solutions, a trivial uniform background soil-water content is assumed. Here, we find new exact analytic and numerical solutions for non-reactive solute transport in non-trivial saturated and unsaturated water flow fields. Furthermore, we investigate the case of adsorption-diffusion which has the added complication of both transport equations being nonlinear of Fokker-Planck type. The nonlinear adsorption-diffusion equation (A.D.E.) is transformed into a class of inhomogeneous nonlinear diffusion equations (I.N.D.E.s). Hidden nonlocal symmetries that seem not to be recorded in the literature are systematically constructed by considering an integrated equation obtained using the general integral variable rather than a system of first order P.D.E.s associated with the concentration and the flux of a conservation law. Reductions for the I.N.D.E. to ordinary differential equations (O.D.E.s) are performed and some invariant solutions are constructed. Also, we obtain solutions for adsorbing solutes in saturated and unsaturated soils.