Symmetry reductions and group-invariant solutions for models arising in water and contaminant transport

Abstract

In this work we consider convection-diffusion equation (CDE) arising in the theory of contamination of water by oil spill. Furthermore, these equations arise in so lute transport and groundwater. Group classification of the one dimensional CDE which depends on time t and space x is performed. Lie point symmetries of the one-dimensional CDE are obtained. Group invariant solutions are constructed using admitted Lie point symmetries and these solutions are used to reduce the CDE to the ordinary differential equations (ODEs), which in most cases are solvable. In cases where a number of symmetries are obtained, we will construct the one-dimensional optimal systems of sub-algebras. The two-dimensional and three dimensional CDE in solute transport with constant dispersion coefficient is considered. In some of these cases, double reduction meth ods will be used. Exact solutions are obtained using the Lie symmetry method in conjunction with the (G0/G)-expansion method and the substitution w(z) = (z0)−1 . To further our studies, we apply the method of potential symmetries to determine group invariant solutions that cannot be obtained using point symmetries. Finally, the non-classical symmetries are obtained and comparison study is done between the results obtained through nonlocal and nonclassical symmetry methods.