3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Classical lie point symmetry analysis of models arising in contaminant transport theory(2014-03-05) Mkhonta, Zwelithini FaneloGroundwater contamination and soil salinisation are a major environmental problem worldwide. Living organisms depend largely on groundwater for their survival and its pollution is of course of major concern. It therefore goes without saying that remedial processes and understanding of the mathematical models that describe contaminant transport is of great importance. The theory of contaminant transport requires understanding of the water ow even at the microscopic level. In this study we focus on macroscopic deterministic models based on di erential equations. Here contaminant will refer to nonreactive contaminant. We aim to calculate Lie point symmetries of the one-dimensional Advection-di usion equation (ADE) for various forms of the di usion coe cient and transport velocity. We aim to employ classical Lie symmetry techniques. Furthermore, reductions will be carried out using the elements of the optimal systems. In concluding, the ADE is analyzed for selected forms of the the di usion coe cient and transport velocity via the potential symmetry method. For the potential symmetries obtained, we investigate the associated invariant solutions.Item Group invariant solutions for contaminant transport in saturated soils under radial uniform water flow background(2013-08-06) Potsane, Moshe MosesThe transport of chemicals through soils to the groundwater or precipitation at the soils surfaces leads to degradation of the resources such as soil fertility, drinking water and so on. Serious consequences may be su ered in the long run. In this dissertation, we consider macroscopic deterministic models de- scribing contaminant transport in saturated soils under uniform radial water ow backgrounds. The arising convection-dispersion equation given in terms of the stream functions is analyzed using classical Lie point symmetries. A number of exotic Lie point symmetries are admitted. Group invariant solu- tions are classi ed according to the elements of the one-dimensional optimal systems. We analyze the group invariant solutions which satisfy some physical boundary conditions. The governing equation describing movements of contaminants under ra- dial water ow background may be given in conserved form. As such, the conserved form of the governing equation may be written as a system of rst order partial di erential equation referred to as an auxiliary system, by an in- troduction of the nonlocal variable. The resulting system of equations admits a number of (local) point symmetries which induce the nonlocal symmetries for the original governing equation. We construct classes of solutions using the admitted genuine nonlocal symmetries, which include the invariant solutions obtained via corresponding point symmetries of the governing equation.