Khalique, C.M.Motsepa, T.Adeyemo, O.D.2020-08-032020-08-032019https://orcid.org/0000-0002-8745-5387http://hdl.handle.net/10394/35448MSc (Applied Mathematics), North-West University, Mafikeng CampusIn this work we examine two nonlinear partial differential equations of fluid mechan-ics. The modified equal-width (MEW) equation, which is used in handling the sim-ulation of a single dimensional wave propagation in nonlinear media with dispersion process, is studied first. We compute the optimal system of one-dimensional subalge-bras and then use it to perform symmetry reductions and obtain group-invariant solutions. Also, we derive conservation laws of the MEW equation using the multiplier approach and the Noether theorem. Secondly we study the generalized nonlinear advection-diffusion equation, which describes the movement of a buoyancy-driven plume in an inclined porous medium. We consider three cases of n and in each case, we construct optimal system of one-dimensional subalgebras using the computed Lie point symmetries and then obtain symmetry reductions and group-invariant solutions based on these optimal systems of one-dimensional subalgebra. In addition, we de-termine the conservation laws of the equation by employing the multiplier approach and the new conservation theorem due to Ibragimov.enThesis