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Fragmentation equations occur naturally in many real world problems, see [ZM85, ZM86, HEL91, CEH91, HGEL96, SLLM00, Ban02, BL03, Ban04, BA06] and references therein. Mathematical study of these equations is mostly concentrated on building existence and uniqueness theories and on qualitative analysis of solutions (shattering), some effort has be done in finding solutions analytically. In this project, we deal with numerical analysis of fragmentation equation with transport. First, we provide some existence results in Banach and Hilbert settings, then we turn to numerical analysis. For this approximation and interpolation theory for generalized Laguerre functions is derived. Using these results we formulate Laguerre pseudospectral method and provide its stability and convergence analysis. The project is concluded with several numerical experiments. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2012.

One of the central difficulties in many models of glacier and ice sheet flow lies in the prescription of boundary conditions at the bed. Often, processes which occur there dominate the evolution of the ice mass as they control the speed at which the ice is able to slide over the bed. In part I of this thesis, we study two complications to classical models of glacier and ice sheet sliding. First, we focus on the effect of cavity formation on the sliding of a glacier over an undeformable, impermeable bed. Our results do not support the widely used sliding law $u_b = C\tau_b^pN^{-q}$, but indicate that $\tau_b/N$ actually decreases with $u_b/N$ at high values of the latter, as suggested previously for simple periodic beds by Fowler (1986). The second problem studied is that of an ice stream whose motion is controlled by bed obstacles with wavelengths comparable to the thickness of ice. By contrast with classical sliding theory for ice of constant viscosity,the bulk flow velocity does not depend linearly on the driving stress. Indeed, the bulk flow velocity may even be a multi-valued function of driving stress and ice thickness. In the second part of the thesis, attention is turned to the formation of drumlins. The viscous till model of Hindmarsh (1998) and Fowler (2000) is analysed in some detail. It is shown that the model does not predict the formation of three-dimensional drumlins, but only that of two-dimensional features, which may be interpreted as Rogen moraines. A non-linear model allows the simulation of the predicted bedforms at finite amplitude. Results obtained indicate that the growth of bedforms invariably leads to cavitation. A model for travelling waves in the presence of cavitation is also developed, which shows that such travelling waves can indeed exist. Their shape is, however, unlike that of real bedforms, with a steep downstream face and no internal stratification. These results indicate that Hindmarsh and Fowler's model is probably not successful at describing the processes which lead to the formation of streamlined subglacial bedforms.

The aim of this thesis is to derive and solve mathematical models for the flow of liquid in a foam. A primary concern is to investigate how so-called `Marangoni stresses' (i.e. surface tension gradients), generated for example by the presence of a surfactant, act to stabilise a foam. We aim to provide the key microscopic components for future foam modelling. We begin by describing in detail the influence of surface tension gradients on a general liquid flow, and various physical mechanisms which can give rise to such gradients. We apply the models thus devised to an experimental configuration designed to investigate Marangoni effects. Next we turn our attention to the flow in the thin liquid films (`lamellae') which make up a foam. Our methodology is to simplify the field equations (e.g. the Navier-Stokes equations for the liquid) and free surface conditions using systematic asymptotic methods. The models so derived explain the `stiffening' effect of surfactants at free surfaces, which extends considerably the lifetime of a foam. Finally, we look at the macroscopic behaviour of foam using an ad-hoc averaging of the thin film models.

The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of $|\psi|^2$, where $\psi$ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the $(n+1)$th state has $n$ zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for $n$ quadruples of complex eigenvalues for the $(n+1)$th state, where a quadruple consists of $\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}$. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.

The aim of this thesis is to investigate the mathematics and modelling of the industrial crimper, perhaps one of the least well understood processes that occurs in the manufacture of artificial fibre. We begin by modelling the process by which the fibre is deformed as it is forced into the industrial crimper. This we investigate by presuming the fibre to behave as an ideal elastica confined in a two dimensional channel. We consider how the arrangement of the fibre changes as more fibre is introduced, and the forces that are required to confine it. Later, we apply the same methods to a fibre confined to a three dimensional channel. After the fibre has under gone a preliminary deformation, a second process known as secondary crimp can occur. This involves the `zig-zagged' material folding over. We model this process in two ways. First as a series of rigid rods joined by elastic hinges, and then as an elastic with a highly oscillatory natural configuration compressed by thrusts at each end. We observe that both models can be expressed in a very similar manner, and both predict that a buckle can occur from a nearly straight initial condition to an arched formation. We also compare the results to experiments performed on the crimped fibre. Throughout much of the process, the configuration of the fibre does not alter. This part of the process we call the block, and model the material in this region in two ways: as a series of springs; and as an isotropic elastic material. We discuss the coupling between the different regions and the process that occurs in the block, and consider both the steady state and stability of the system.

Frost heave describes the phenomenon whereby soil freezing causes upwards surface motion due to the action of capillary suction imbibing water from the unfrozen region below. The expansion of water on freezing is a small part of the overall surface heave and it is the flow of water towards the freezing front which is largely responsible for the uplift. In this thesis, we analyse a model of frost heave due to Miller (1972, 1978) which is referred to as `secondary frost heave'. Secondary frost heave is characterised by the existence of a `partially frozen zone', underlying the frozen soil, in which ice and water coexist in the pore space. In the first part of the thesis we follow earlier work of Fowler, Krantz and Noon where we show that the Miller model for incompressible soils can be dramatically simplified. The second part of the thesis then uses this simplification procedure to develop simplified models for saline and compressible soils. In the latter case, the development of the theory leads to the consideration of non-equilibrium soil consolidation theory and the formation of segregated massive ice within permafrost. The final part of the thesis extends the simplified Miller model to the analysis of differential frost heave and the formation of patterned ground (e.g. earth hummocks and stone circles). We show that an instability mechanism exists which provides a plausible theory for the formation of these types of patterned ground.