3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item An analysis of symmetries and conservation laws of nonlinear partial differential equations arising from Burgers’ hierarchy(2020) Obaidullah, UsaamahWe investigate the nonlinear evolutionary partial differential equations (PDEs) derived from Burgers’ hierarchy and give the exact solution of the complete hierarchy. The conservation laws of the hierarchy are studied and we proceed to establish the general nth conservation law. A transformation is used to render the hierarchy to a hierarchy of nonlinear ordinary differential equations (ODEs). These expressions are then linearised. Ultimately we give a novel exact solution of the entire Burgers’ hierarchy, that is, for all values of n. Several members of the hierarchy are solved, and the graphical counterparts of their solutions are provided to illustrate the applicability of our formula. Next we extend our study to the hierarchy of ODEs linked to this hierarchy. One-parameter Lie group of transformations that leave the ODEs invariant are constructed, from which it is established that these symmetries arise from the (n+ 1) complex roots of a certain polynomial. This gives us a formula to solve the ODE expressions, and finally we show how a more general exact solution of the complete hierarchy is obtained from this resultItem Symmetry structures and conserved forms of systems of pdes(2019) Alqurashi, Bader MutairWe will study the symmetry, invariance properties and conservation laws of partial dif ferential equations (pdes) that arise in a number of situations in mathematical physics. These will be range from Image Processing and noise removal algorithms to Timoshenko beam systems. Furthermore, we will study the invariance properties and approximate conservation laws of some nonlinear Schro¨dinger equation with PT-symmetric potentials with inhomogeneous nonlinearity and some nonlinear Schro¨dinger equation involving a spatially extended system consisting of two coupled elements.Item Modelling of gas recovery from South African shale reservoirs(2019-09-09) Qwabe, Thembinkosi SabeloMathematical models for a triple-porosity continuum describing the distribution of pressure-the driving force for gas flow during gas production-were developed on the basis of the continuity equation from first principles, focusing on the South African shales of the Karoo Basin. The triple-porosity continuum incorporated the matrix, the natural fracture network, and the hydraulic fracture network; the matrix and the natural fracture network systems were considered two dimensional, while the hydraulic fracture system was considered one dimensional. The three developed mathematical models are in the form of the general diffusion or heat equation, thus the numerical Finite difference method (FDM) was employed in solving the triple-porosity model. MATLAB software was used to develop and solve the FDM simulation or algorithm of each system of the triple-porosity model; results for each system are presented and discussed in chapter 6 of the research report. The results generated agree with models previously developed and validated with their respective field data, substantiating the reliability of the model developed in this research report. Currently the South Africa’s Barnett shale does not have field data, hence the field data from literature was used to validate the developed model. The model is thus a guide onto describing the gas flow behaviour for the South African shale gas reservoirs, and not specifically for the Barnett shale.Item Modelling of gas recovery from unconventional reservoir shale gas(2019) Khari, AndileIt is believed that petroleum gas is stored as free gas in natural fractures, free gas in pores, adsorbed gas and dissolved gas in kerogen bulk in the shale. Hydraulic Fracking is used to promote free gas flow to production well, but adsorbed and dissolved gas is not recovered and usually ignored. This work looks into modifying existing gas flow model, by including mechanisms that can promote and contribute to shale gas production recovery during reservoir depletion. The developed model includes non-Darcy flow which is suitable for high-velocity gas flow. Model equation was simulated with the help of MATLAB to solve the partial derivative equation to achieve a shale gas production a 3D profile of pressure vs time vs distance. The model results were seen behaving similarly to available developed models, whereby pressure is initially increasing and then decrease overtime. The initial increase in pressure is due to the free gas available in natural fractures and micro-fracture in the matrix and is produced first causing pressure to increase. During production, overtime free gas in these natural fractures and micro-fracture gets depleted, causing pressure to decrease and approach critical desorption pressure over time. The free gas in the matrix nanopores feed these depleted fracture networks, the kerogen nanopores is in turn fed by adsorbed and dissolved gas on kerogen inside the nanopores surface which take place at later stage of production. The developed model shows similar gas recovery production behaviour with the laboratory results, which proves that the proposed model can be used to predict the production profileItem Modelling of gas recovery from South African shale reservoirs (focusing on the KWV-1 bore hole in the Eastern Cape Province)(2018) Makoloane, NkhabuThe main aim of the study was to develop mathematical flow model of the shale gas at the Karoo Basin of South Africa (SA). The model development incorporates three systems (phases) to form a triple continuum flow model, the phases include matrix (m), natural (NF) and hydraulic fracture (HF). The model was developed from the continuity equation, and the general equations were formed. (0.05������ ���� = 3.90087 × 10−15 ��2���� ����2 + 3.90087 × 10−15 ��2���� ����2 − 1.95043 × 10−16(20 × 106 − ������), 0.01 �������� ���� = 2.00 × 10−15(20 × 106 − ������) − 2.00 × 10−9(20 × 106 − ������) + �� ���� [7.80 × 10−5 �������� ���� ] + �� ���� [7.80 × 10−5 �������� ���� ] �� ���� [0.1248269 �������� ���� ] + 0.1248269(20 × 106 − ������)− 4.98 × 10−4 = �������� ���� The model was solved using numerical method technique known as Finite Difference Method (FDM). For each phase a computer program MATLAB was used to plot the pressure gradient. Hydraulic pressure gradient fractures propagate between the distance of 100m and 500m. The model was verified using the data of Barnett Shale. Sensitivity analysis was also performed on the hydraulic permeability, drainage radius and the initial pressure of the reservoir.Item Symmetry reductions and group-invariant solutions for models arising in water and contaminant transport(2018) Ntsime, B.PIn 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.Item The application of non-linear partial differential equations for the removal of noise in audio signal processing(2017) Shipton, Jarrod JayThis work explores a new method of applying partial di erential equations to audio signal processing, particularly that of noise removal. Two methods are explored and compared to the method of noise removal used in the free software Audacity(R). The rst of these methods uses a non-linear variation of the di usion equation in two dimensions, coupled with a non-linear sink/source term, in order to lter the imaginary and real components of an array of overlapping windows of the signal's Fourier transform. The second model is that of a non-linear di usion function applied to the magnitude of the Fourier transform in order to estimate the noise power spectrum to be used in a spectral subtraction noise removal technique. The technique in this work features nite di erence methods to approximate the solutions of each of the models.Item Symmetry analysis and invariant properties of some partial differential equations(2017) Mathebula, AgreementThis dissertation contains evolutionary partial differential equations (PDEs). The PDEs are used to investigate ecological phenomena. The main goal is to determine Lie point symmetries, perform Lie reduction, obtain analytical solutions and visualize the solutions in 3D plots using the help of Mathematica. Drift diffusion, biased diffusion and the Kierstead, Slobodkin and Skellam (KiSS) models arising in population ecology are discussed. The importance of these PDEs in ecology is to analyse the movements of organisms and their long-term existence especially in heterogeneous environments.