In this work, we studied heat and mass transfer in a nanofluid flow over a stretching sheet. Fluid flow in different flow geometries was studied and a co-ordinate transformation was used to transform the governing equations into non-dimensional non-similar boundary layer equations. These equations were then solved numerically using both established and recent techniques such as the spectral relaxation and spectral quasi-linearization methods. Numerical solutions for the heat transfer, mass transfer and skin friction coefficients have been presented for different system parameters, such as heat generation, Soret and Dufour effects, chemical reaction, thermal radiation influence, the local Grashof number, Prandtl number, Eckert number, Hartmann number and the Schmidt number. The dependency of the skin friction, heat and mass transfer coefficients on these parameters has been quantified and discussed. The accuracy, and validity of the spectral relaxation and spectral quasi-linearization methods has been established.

Approaches to nding solutions to di erential equations are usually ad hoc. One of the more successful methods is that of group theory, due to Sophus Lie. In the case of ordinary di erential equations, the subsequent symmetries obtained allow one to reduce the order of the equation. In the case of partial di erential equations, the symmetries are used to nd (particular) group invariant solutions by reducing the number of variables in the original equation. In the latter case, these solutions are particularly popular in applications as they are often the only physically signi cant ones obtainable. As a result, it is now becoming traditional to apply this symmetry method to nd solutions to di erential equations in a systematic manner. Based upon the Lie algebra of symmetries of the equation, we expect a certain number of symmetries after the reductions. However, it has become increasingly observed that, after reduction, more symmetries than expected are often obtained. These are called Hidden Symmetries and they provide new routes for further reduction. The idea of our research is to give an overview of this phenomenon. In particular, we investigate the possible origins of these symmetries. We show that they manifest themselves as nonlocal symmetries (or potential symmetries), contact symmetries or nonlocal contact symmetries of the original equation as well as point symmetries of another equation of same order. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

We investigate the relationship between the Dynamical Systems analysis and the Lie Symmetry analysis of ordinary differential equations. We undertake this investigation by looking at a relativistic model of self-gravitating charged fluids. Specifically we look at the impact of specific parameters obtained from Lie Symmetries analysis on the qualitative behaviour of the model. Steady states, stability and possible bifurcations are explored. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

Thesis (PhD)--Stellenbosch University, 1975. / ENGLISH ABSTRACT: Some problems of applied mathematics, for instance in the fields of aerodynamics or electron optics, involve certain singular integrals which do not exist classically. The problems can, however, be solved pLovided that such integrals are interpreted as finite-part integrals. Although the concept of a finite-part integral has existed for about fifty years, it was possible to define it rigorously only by means of distribution theory, developed about twenty-five years ago. But, to the best of our knowledge, no quadrature formula for the numerical eva= luation of finite-part integrals ha~ been given in the literature. The main concern of this thesis is the study and discussion of.two kinds of quadrature formulae for evaluating finite-part integrals in= volving an algebraic singularity. Apart from a historical introduction, the first chapter contains some physical examples of finite-part integrals and their definition based on distribution theory. The second chapter treats the most im= portant properties of finite-part integrals; in particular we study their behaviour under the most common rules for ordinary integrals. In chapters three and four we derive a quadrature formula for equispaced stations and one which is optimal in the sense of the Gauss-type quadra= ture. In connection with the latter formula, we also study a new class of orthogonal polynomials. In the fifth and.last chapter we give a derivative-free error bound for the equispaced quadrature formula. The error quantities which are independent of the integrand were computed for the equispaced quadrature formula and are also given. In the case of some examples, we compare the computed error bounds with the actual errors. ~esides this theoretical investigation df finite-part integrals, we also computed - for several orders of the algebraic singularity the coefficients for both of the aforesaid quadrature formulae, in which the number of stations ranges from three up to twenty. In the case of the equispaced quadrature fortnu1a,we give the weights and - for int~ger order of the singularity - the coefficients for a numerical derivative of the integrand function. For the Gauss-type quadrature, we give the stations, the corresponding weights and the coefficients of the orthogonal polynomials. These data are being published in a separate report [18] which also contains detailed instructions on the use of the tables.

In this dissertation we investigate free convective heat and mass transfer in micropolar fluid flow past a moving semi-infinite vertical porous plate in the presence of a magnetic field. The aim of this study was to use recent semi-numerical methods such as the successive linearisation method and the spectral-homotopy analysis method to study the effects of viscous heating and the effects of different fluid parameters. The governing boundary layer equations for linear momentum, angular momentum (microrotation), temperature and concentration profiles are transformed to a system of ordinary differential equations and solved using the successive linearisation method and the spectral-homotopy analysis method. The accuracy of the solutions was determined by comparison with numerical approximations obtained using the Matlab bvp4c solver. The influences of the micropolar parameter, Darcy number, Prandtl number, Schmidt number, magnetic parameter, heat absorption parameter, Soret and Dufour numbers, local Reynolds number and Grashof number on velocity, microrotation, temperature and concentration profiles were determined. The results obtained are presented graphically and in tabular form. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

The Frobenius group is an example of a split extension. In this dissertation we study and describe the properties and structure of the group. We also describe the properties and structure of the kernel and complement, two non-trivial subgroups of every Frobenius group. Examples of Frobenius groups are included and we also describe the characters of the group. Finally we construct the Frobenius group 292 : SL(2, 5) and then compute it's Fischer matrices and character table. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

Lie symmetry group methods provide a useful tool for the analysis of differential equations in a variety of areas in physics and applied mathematics. The nature of symmetry is that it provides information on properties which remain invariant under transformation. In differential equations this invariance provides a route toward complete integrations, reductions, linearisations and analytical solutions which can evade standard techniques of analysis. In this thesis we study two problems in quantum mechanics from a symmetry perspective: We consider for pedagogical purposes the linear time dependent Schrodinger equation in a potential and provide a symmetry analysis of the resulting equations. Thereafter, as an original contribution, we study the group theoretic properties of the density matrix equation for the quantum Brownian motion of a free particle interacting with a bath of harmonic oscillators. We provide a number of canonical reductions of the system to equations of reduced dimensionality as well as several complete integrations. / Thesis (M.Sc.) - University of KwaZulu-Natal, Westville, 2008.

A steady state two-dimensional mixed convection problem in an air filled square unit cavity has been numerically investigated. Two different cases of heating are investigated and compared. In the first case, the bottom wall was uniformly heated, the side walls were linearly heated and the top moving wall was heated sinusoidally. The second case differed from the first in that the side walls were instead uniformly cooled. This investigation is an extension of the work by Basak et al. [6, 7] who investigated mixed convection in a square cavity with similar boundary conditions to the cases listed above with the exception of the top wall which was well insulated. In this dissertation, their work is extended to include a sinusoidally heated top wall. The nonlinear coupled equations are solved using the Penalty Galerkin Finite Element Method. Stream function and isotherm results are found for various values of the Reynolds number and the Grashof number. The strength of the circulation is seen to increase with increasing Grashof number and to decrease with increasing Reynolds number for both cases of heating. A comparison is made between the stream function and isotherm results for the two cases. The results for the rate of heat transfer in terms of the Nusselt number are discussed. Both local and average Nusselt number results are presented and discussed. The average Nusselt number is found using Simpson's 1/3rd rule. The rate of heat transfer is found to be higher at all four walls for the case of cooled side walls than that of linearly heated side walls. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2010.

In this thesis we study double-diffusive convection and cross-diffusion effects in flow through porous media. Fluid flows in various flow geometries are investigated and the governing equations are solved analytically and numerically using established and recent techniques such as the Keller-box method, the spectral-homotopy analysis method and the successive linearisation method. The effects of the governing parameters such as the Soret, Dufour, Lewis, Rayleigh and the Peclet numbers and the buoyancy ratio on the fluid properties, and heat and mass transfer at the surface are determined. The accuracy, computational efficiency and validity of the new methods is established. This study consists of five published and one submitted paper whose central theme is the study of double-diffusive convection in porous media. A secondary theme is the application of recent numerical semi-numerical methods in the solution of nonlinear boundary value problems, particularly those that arise in the study of fluid flow problems. Paper 1. An investigation of the quiescent state in a Maxwell fluid with double-diffusive convection in porous media using linear stability analysis is presented. The fluid motion is modeled using the modified Darcy-Brinkman law. The critical Darcy- Rayleigh numbers for the onset of convection are obtained and numerical simulations carried out to show the effects of the Soret and Dufour parameters on the critical Darcy-Rayleigh numbers. For some limiting cases, known results in the literature are recovered. Paper 2. We present an investigation of heat and mass transfer in a micropolar fluid with cross-diffusion effects. Approximate series solutions of the governing non-linear differential equations are obtained using the homotopy analysis method (HAM). A comparison is made between the results obtained using the HAM and the numerical results obtained using the Matlab bvp4c numerical routine. Paper 3. The spectral homotopy analysis method (SHAM) as a new improved version of the homotopy analysis method is introduced. The new technique is used to solve the MHD Jeffery-Hamel problem for a convergent or divergent channel. We show that the SHAM improves the applicability of the HAM by removing the restrictions associated with the HAM as well as accelerating the convergence rate. Paper 4. We present a study of free and forced convection from an inverted cone in porous media with diffusion-thermo and thermo-diffusion effects. The highly nonlinear governing equations are solved using a novel successive linearisation method (SLM). This method combines a non-perturbation technique with the Chebyshev spectral collection method to produce an algorithm with accelerated and assured convergence. Comparison of the results obtained using the SLM, the Runge-Kutta together with a shooting method and the Matlab bvp4c numerical routine show the accuracy and computational efficiency of the SLM. Paper 5. Here we study cross-diffusion effects and convection from inverted smooth and wavy cones. In the case of a smooth cone, the highly non-linear governing equations are solved using the successive linearisation method (SLM), a shooting method together with a Runge-Kutta of order four and the Matlab bvp4c numerical routine. In the case of the wavy cone the governing equations are solved using the Keller-box method. Paper 6. We examine the problem of mixed convection, heat and mass transfer along a semi-infinite plate in a fluid saturated porous medium subject to cross-diffusion and radiative heat transfer. The governing equations for the conservation of momentum, heat and solute concentration transfer are solved using the successive linearisation method, the Keller-box technique and the Matlab bvp4c numerical routine. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

With the classification of finite simple groups having been completed in 1981, recent work in group theory has involved the study of the structures of simple groups. The character tables of maximal subgroups of simple groups give substantive information about these groups. Most of the maximal subgroups of simple groups are of extension type. Some of the maximal subgroups of simple groups contain constituents of the generalized symmetric groups. Here we shall be interested in discussing such groups which we may call groups associated with the generalized symmetric groups. There are several well developed methods for calculating the character tables of group extensions. However Fischer [17] has given an effective method for calculating the character tables of some group extensions including the generalized symmetric group B (m, n). Actually work on the characters of wreath products with permutation groups dates back to Specht's work [61], through the works of Osima [49] and Kerber [33]. And more recently other people have worked on characters of wreath products with symmetric groups, these amongst others include Darafshesh and Iranmanesh [14], List and Mahmoud [36], Puttaswamiah [52], Read [55, 56], Saeed-Ul-Islam [59] and Stembridge [64]. It is well known that the character table of the generalized symmetric group B(m, n), where m and n are positive integers, can be constructed in GAP [22] with B(m, n) considered as the wreath product of the cyclic group Zm of order m with the symmetric group Sn' For example Pfeiffer [50] has given programmes which compute the character tables of wreath products with symmetric groups in GAP. However it may be necessary to obtain the partial character table of a group in hand rather than its complete character table. Further due to limited workspace in GAP, the wreath product method can only be used to compute character tables of B(m, n) for small values of m and n. It is for these reasons amongst others that Fischer's method is sometimes used to construct the character tables of such groups. groups B(2, 6) and B(3, 5) of orders 46080 and 29160 is done here. We have also used Programme 5.2.4 to construct the Fischer-Clifford matrices of the groups B(2, 12) and B(4, 5) of orders 222 x 35 X 52 X 7 x 11 and 213 x 3 x 5 respectively. Due to lack of space here we have given the Fischer-Clifford matrices of B(2, 12) and B(4,5) on the compact disk submitted with this thesis. However note that these matrices are the equivalent form of the Fischer-Clifford matrices of B(2, 12) and B(4,5). In [35] R.J. List has presented a method for constructing the Fischer-Clifford matrices of group extensions of an irreducible constituent of the elementary abelian group 2n by a symmetric group. The other aim of our work is to adapt the combinatorial method in [5] to the construction of the Fischer-Clifford matrices of some group extensions associated with B(m, n), using a similar method as the one used in [35]. Examples are given on the application of this adaptation to some groups associated with the groups B(2, 6), B(3,3) and B(3, 5). In this thesis we have constructed the character tables of the groups B(2, 6) and B(3,5) and some group extensions associated with these two groups and B(3, 3). We have also constructed the character tables of the groups B(2, 12) and B(4, 5) in our work, these character tables are given on the compact disk submitted with this thesis. The correctness of all the character tables constructed in this thesis has been tested in GAP. The main working programmes (Programme 2.2.3, Programme 3.1.9, Programme 3.1.10, Programme 5.2.1, Programme 5.2.4 and Programme 5.2.2) are given on the compact disk submitted with this thesis. It is anticipated that with further improvements, a number of the programmes given here will be incorporated into GAP. Indeed with further research work the programmes given here should lead to an alternative programme for computing the character table of B(m, n). / Thesis (Ph.D.)- University of KwaZulu-Natal, Pietermaritzburg, 2005.