In this dissertation we consider the two dimensional flow of an incompressible and electrically conducting second grade fluid past a vertical porous plate with constant suction. The flow is permeated by a uniform transverse magnetic field. The aim of this study is to use the multi-parameter perturbation technique to study the effects of Eckert numbers on the flow of a pulsatile second grade fluid along a vertical plate. We further aim to investigate the effects of other fluid and physical parameters such as the Prandtl numbers, Hartmann numbers, viscoelastic parameter, angular frequency and suction velocity on boundary layer velocity, temperature, skin friction and the rate of heat transfer. Similarity transformations are used to reduce the governing partial differential equations to ordinary differential equations. We used perturbation methods to solve the coupled ordinary differential equations for zero Eckert number and the multiparameter perturbation technique to solve the coupled ordinary differential equations for small viscoelastic parameters and Eckert numbers. It is found that increasing the Eckert number or the viscoelastic parameter enhances the boundary layer velocity while reducing the temperature, the rate of heat transfer and the skin-friction. The results for the boundary layer velocity and the temperature are presented graphically and discussed. The results for the rate of heat transfer in terms of the Nusselt number and the skin friction are tabulated and discussed. A good agreement is found between these results and other published research. The comparison between the results for zero Eckert numbers and small Eckert numbers is also presented graphically and discussed. / Thesis (M.Sc.) - University of KwaZulu-Natal, Pietermaritzburg, 2009.

The current state of information concerning the classical model of deterministic, age-dependent population dynamics - the McKendrick von Foerster equation - is overviewed. This model and the related Renewal equation are derived and the parameters involved in both are elaborated upon. Fundamental theorems concerning existence, uniqueness and boundedness of solutions are outlined. A necessary and sufficient condition concerning the stability of equilibrium age-distributions is rederived along different lines. Attention is then given to generalizations of the McKendrick-von Foerster model that have arisen from the inclusion of density- dependence into the parameters of the system; the inclusion of harvesting terms; and the extension of the model to describe the dynamics of a two-sex population. A technique which reduces the model, under certain conditions on the mortality and fertility functions, to a system of ordinary differential equations is discussed and applied to specific biochemical population models. Emphasis here is on the possible existence of stable limit cycles.The Kolmogorov system of ordinary differential equations and its use in describing the dynamics of predator-prey systems is examined. The Kolmogorov theorem is applied as a simple alternative to a lengthy algorithm for determining whether limit cycles are stable. Age-dependence is incorporated into this system by means of a McKendrick - von Foerster equation and the effects on the system of different patterns of age-selective predation are demonstrated. Finally, brief mention is made of recent work concerning the use of the McKendrick - von Foerster equation to describe the dynamics of both predator and prey. A synthesis of the theory and results of a large number of papers is sought and areas valuable to further research are pointed out. / Thesis (M.Sc.)-University of Natal, Durban, 1988.

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In this thesis we derive and solve equations governing the flow of slender threads and sheets of viscous fluid. Our method is to solve the Navier-Stokes equations and free surface conditions in the form of asymptotic expansions in powers of the inverse aspect ratio of the fluid, i.e. the ratio of a typical thickness to a typical length. In the first chapter we describe some of the many industrial processes in which such flows are important, and summarise some of the related work which has been carried out by other authors. We introduce the basic asymptotic methods which are employed throughout this thesis in the second chapter, while deriving models for two-dimensional viscous sheets and axisymmetric viscous fibres. In chapter 3 we show that when these equations govern the straightening or buckling of a curved viscous sheet, simplification may be made via the use of a suitable short timescale. In the following four chapters, we derive models for nonaxisymmetric viscous fibres and fully three-dimensional viscous sheets; for each we consider separately the cases where the dimensionless curvature is small and where the dimensionless curvature is of order one. We find that the models which result bear a marked similarity to the theories of elastic rods, plates and shells. In chapter 8 we explain in some detail why the Trouton ratio - the ratio between the extensional viscosity and the shear viscosity - is 3 for a slender viscous fibre and 4 for a slender viscous sheet. We draw our conclusions and suggest further work in the final chapter.

Many physics and engineering problems are modeled by differential equations. In many instances these equations are nonlinear and exact solutions are difficult to obtain. Numerical schemes are often used to find approximate solutions. However, numerical solutions do not describe the qualitative behaviour of mechanical systems and are insufficient in determining the general properties of certain systems of equations. The need for analytical methods is self-evident and major developments were seen in the 1990’s. With the aid of faster processing equipment today, we are able to compute analytical solutions to highly nonlinear equations that are more accurate than numerical solutions. In this study we discuss solutions to nonlinear partial differential equations with focus on non-perturbation analytical methods. The non-perturbation methods of choice are the homotopy analysis method (HAM) developed by Shijun Liao and the variational iteration method (VIM) developed by Ji-Huan He. The aim is to compare the solutions obtained by these modern day analytical methods against each other focusing on accuracy, convergence and computational efficiency. The methods were applied to three test problems, namely, the heat equation, Burgers equation and the Bratu equation. The solutions were compared against both the exact results as well as solutions generated using the finite difference method, in some cases. The results obtained show that the HAM successfully produces solutions which are accurate, faster converging and requires less computational resources than the VIM. However, the VIM still provides accurate solutions that are also in good agreement with the closed form solutions of the test problems. The FDM also produced good results which were used as a further comparison to the analytical solutions. The findings of this study is in agreement with those published in the literature. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

This thesis explains the theoretical background of stochastic differential equations in one dimension. We also show how to solve such differential equations using strong It o-Taylor expansion schemes over large time grids. We also attempt to solve a problem regarding a specific approximation of a stochastic integral for which there is no explicit solution. This approximation, which utilizes the distribution of this particular stochastic integral, gives the wrong order of convergence when performing a grid convergence study. We use numerical integration of the stochastic integral as an alternative approximation, which is correct with regards to convergence.

In this thesis we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.

The purpose of this thesis is to determine the most effective parallel algorithm for the solution of the parabolic differential equations characteristic of diffusion problems. The primary aim is to apply the chosen algorithm to obtain solutions to the equations governing the operation of membrane-covered oxygen sensors, known as Clark electrodes, which are used for monitoring the oxygen concentration of blood. The boundary conditions of this problem require the development of a singularity correction technique. A brief history of electrochemical sensors leading to the development of the Clark electrode is given, together with the two-dimensional equations and boundary conditions governing its operation. A locally valid series expansion is derived to take care of the boundary singularity, together with a robust method of matching this to the finite difference approximation. Parallel implementations of three representative numerical algorithms applied to a simple model problem are compared by extending Leland's parallel effectiveness model. The chosen parallel algorithm is combined with the singularity correction to obtain a solution to the Clark electrode problem. Numerical experiments show this solution to achieve the required accuracy. Previous one-dimensional models of the Clark electrode are shown to be inadequate before the two-dimensional model is used to examine the variation of operation with design. The understanding gained allows us to demonstrate the advantages of pulse amperometry over steady-state techniques, and to suggest the most appropriate method and design for use in in vivo clinical monitoring.