We consider bounded weak solutions u of a degenerate parabolic-hyperbolic equation defined in a subset [mathematical symbols]. We define strong notion of trace at the boundary [mathematical symbols] reached by L¹ convergence for a large class of functionals of u. Such functionals depend on the flux function of the degenerate parabolic-hyperbolic equation and on the boundary. We also prove the well-posedness of the entropy solution for scalar conservation laws with a strong boundary condition with the above trace result as applications. / text

PhD (Applied Mathematics) / Department of Mathematics and Applied Mathematics / The work presented in this thesis is the application of the recently introduced numerical techniques, namely the spectral quasi-linearization method (SQLM) and the bivariate spectral quasi-linearization method (BSQLM), in solving problems arising in fluid flow. Firstly, we use the SQLM to solve the highly non-linear one dimensional Bratu problem. The results obtained are compared with exact solution and previously published results using the B-spline method, Picard’s Green’s Embedded Method and the iterative finite difference method. The results obtained show that the SQLM is highly accurate and computationally efficient. Secondly, we use the bivariate spectral quasi-linearization method to solve the two dimensional Bratu problem. Since the exact solution of the two-dimensional Bratu problem is unknown, the results obtained are compared with those previously published results using the finite difference method and the weighted residual method. Thirdly, we use the BSQLM to study numerically the boundary layer flow of a third grade non-Newtonian fluid past a vertical porous plate. We use the Jeffrey fluid as a typical fluid which shows non-Newtonian characteristics. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations which are then solved using BSQLM. The influence of some thermo-physical parameters namely, the ratio relaxation to retardation times parameter, Prandtl number, Schmidt number and the Deborah number is investigated. Also investigated is the influence of the ratio of relaxation to retardation times, Schmidt number and the Prandtl number on the skin friction, heat transfer rate and the mass transfer rate. The results obtained show that increasing the Schmidt number decelerates the fluid flow, reduces the skin friction, heat and mass transfer rates and strongly depresses the fluid concentration whilst the temperature is increased. The fluid velocity, the skin friction, heat and mass transfer rates are increased with increasing values of the relaxation to retardation parameter whilst the fluid temperature and concentration are reduced. Using the the solution based errors, it was shown that the BSQLM converges to the solution only after 5 iterations. The residual error infinity norms showed that BSQLM is very accurate by giving an error of order of 10−4 within 5 iterations. Lastly we propose a model of the non-Newtonian fluid flow past a vertical porous plate in the presence of thermal radiation and chemical reaction. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations. The BSQLM is used to solve the system of equations. We investigate the influence of the ratio of relaxation to retardation parameter, Schmidt number, Prandtl number, thermal radiation parameter, chemical reaction iv parameter, Nusselt number, Sherwood number, local skin fiction coefficient on the fluid concentration, fluid temperature as well as the fluid velocity. From the study, it is noted that the fluid flow velocity, the local skin friction coefficient, heat and mass transfer rate are increased with increasing ratio of relaxation to retardation times parameter whilst the fluid concentration is depressed. Increasing the Prandtl number causes a reduction in the velocity and temperature of the fluid whilst the concentration is increased. Also, the local skin friction coefficient and the mass transfer rates are depressed with an increase in the Prandtl number. An increase in the chemical reaction parameter decreases the fluid velocity, temperature and the concentration. Increasing the thermal radiation parameter has an effect of decelerating the fluid flow whilst the temperature and the concentration are slightly enhanced. The infinity norms were used to show that the method converges fast. The method converges to the solution within 5 iterations. The accuracy of the solution is checked using residual errors of the functions f, and . The errors show that the BSQLM is accurate, giving errors of less than 10−4, 10−7 and 10−8 for f, and , respectively, within 5 iterations. / NRF

Scientists' ability to generate and store simulation results is outpacing their ability to analyze them via ad hoc programs. We observe that these programs exhibit an algebraic structure that can be used to facilitate reasoning and improve performance. In this dissertation, we present a formal data model that exposes this algebraic structure, then implement the model, evaluate it, and use it to express, optimize, and reason about data transformations in a variety of scientific domains. Simulation results are defined over a logical grid structure that allows a continuous domain to be represented discretely in the computer. Existing approaches for manipulating these gridded datasets are incomplete. The performance of SQL queries that manipulate large numeric datasets is not competitive with that of specialized tools, and the up-front effort required to deploy a relational database makes them unpopular for dynamic scientific applications. Tools for processing multidimensional arrays can only capture regular, rectilinear grids. Visualization libraries accommodate arbitrary grids, but no algebra has been developed to simplify their use and afford optimization. Further, these libraries are data dependent—physical changes to data characteristics break user programs. We adopt the grid as a first-class citizen, separating topology from geometry and separating structure from data. Our model is agnostic with respect to dimension, uniformly capturing, for example, particle trajectories (1-D), sea-surface temperatures (2-D), and blood flow in the heart (3-D). Equipped with data, a grid becomes a gridfield. We provide operators for constructing, transforming, and aggregating gridfields that admit algebraic laws useful for optimization. We implement the model by analyzing several candidate data structures and incorporating their best features. We then show how to deploy gridfields in practice by injecting the model as middleware between heterogeneous, ad hoc file formats and a popular visualization library. In this dissertation, we define, develop, implement, evaluate and deploy a model of gridded datasets that accommodates a variety of complex grid structures and a variety of complex data products. We evaluate the applicability and performance of the model using datasets from oceanography, seismology, and medicine and conclude that our model-driven approach offers significant advantages over the status quo.

LD2668 .T4 1942 F3 / Master of Science

In Chinese culture, certain digits are considered lucky and others unlucky. This thesis evaluates how numerical superstition affects financial decision-making in the Chinese A-share IPO market for the period between 2003-2015. Evidence has been found that suggests that numerical superstition influences the initial return on the issuing day of A-share IPOs on the Shanghai exchange. On this exchange newly listed firms with the unlucky number 4 and lucky numbers 6 and 8 in their ticker are initially traded at a discount. A superstition effect for the lucky numbers 6 and 8 dissipates after the first trading day but remains visible after the IPO for the unlucky number 4 and disappears within a month. The Shenzhen exchange showed no effects of numerical superstition on the initial return of the first trading date. The additional regression results indicate that after one month and onwards, having an unlucky number in a ticker has a negative influence on IPO underpricing . After the 3rd and 6th month the lucky number 6 is has a significant negative impact on stock return.

<p>Ocean-based wireless sensor networks serve many important purposes ranging from tsunami early warning to anti-submarine warfare. Developing energy harvesting devices that make these networks self-sufficient allows for reduced maintenance cost and greater reliability. Many methods exist for powering these devices, including internal batteries, photovoltaic cells and thermoelectric generators, but the most reliable method, if realized, would be to power these devices with an internal kinetic energy harvester capable of reliably converting wave motion into electrical power. Designing such a device is a challenge, as the ocean excitation environment is characterized by shifting frequencies across a relatively wide bandwidth. As such, traditional linear kinetic energy harvesting designs are not capable of reliably generating power. Instead, a nonlinear device is better suited to the job, and the task of this dissertation is to investigate the behaviors of devices that could be employed to this end.</p><p>This dissertation is motivated by the design and analysis of an ocean energy harvester based on a horizontal pendulum system. In the course of investigating the dynamics of this system, several discoveries related to other energy harvesting systems were made and are also reported herein. It is found that the most reliable method of characterizing the behaviors of a nonlinear energy harvesting device in the characteristically random forcing environment of the ocean is to utilize statistical methods to inform the design of a functional device. It is discovered that a horizontal pendulum-like device could serve as an energy harvesting mechanism in small self-</p><p>sufficient wireless sensor buoys if properly designed and if the proper transduction mechanisms are designed and employed to convert the mechanical energy of the device into electrical power.</p> / Dissertation

A new method for generating initial data for simulations of neutron stars in binary systems. The construction of physically relevant initial data is crucial to accurate assessment of gravitational wave signals relative to theoretical predictions. This method builds upon the Bowen-York curvature for puncture black holes. This data is evolved and compared against simulations in the literature with respect to orbital eccentricity, merger and collapse times, and emitted energy and angular momentum. The data exhibits some defects, including large central density oscillations in stars and center of mass drift in unequal-mass systems. Some approaches for improvements in potential future work are discussed.

Anthropogenic noise in the sea is now classed as a pollutant alongside chemical pollution and marine litter in accordance with the Marine Strategy Framework Directive. Noise from shipping is a major contributor to the ambient noise levels in the ocean, particularly at low (&lt;300Hz) frequencies. The properties of sound propagation in shallow waters are highly influenced by the marine physical environment. Ocean modelling plays an important role in underwater noise studies since it can provide high resolution water column parameters over large geographic areas. This study investigates the noise patterns and their temporal variations in the Celtic Sea by using a coupled ocean model (POLCOMS) and an acoustic model (HARCAM). A method to predict noise exposure experienced by marine animals is then developed, following an application for diving seals. The ocean model is applied in the Celtic Sea to provide high-resolution 3D hourly temperature and salinity fields for the acoustic model. The model is validated against in-situ and satellite observations, giving high skills to simulate the water column structures. Sensitivity studies of modelled results to different atmospheric forcing are carried out in order to improve the accuracy of the model. The results show that the modelled sea surface temperature, stratification and water column structures are highly sensitive to the choice of surface forcing, especially in the summer time. The increase in resolution of surface forcing does not necessarily lead to more accurate results. The tidally frontal position is, however, insensitive to the forcing. The variability of noise propagation is studied using the coupled model, demonstrating high dependence on oceanographic conditions, geographic location of sound source and its depth. In summer, when the source of sound is on the inshore side of the bottom front, the sound energy is mostly concentrated in the near-bottom layer. In winter, the sound from the same source is distributed more evenly in the vertical. When the source is on the seaward side of the front, the sound level from a shallow source is nearly uniform in the vertical and the transmission loss is significantly greater (~16dB at 40km distance) in summer than in winter. In contrast, sound energy from a deep source is trapped in the bottom cold water, leading to a much lower transmission loss (~20dB) in summer than in winter. Note that ~10dB fluctuation of sound energy is found during the deterioration of the thermocline in late autumn. Shallow sources (e.g. ships) are sensitive to the surface heat flux as it changes significantly the vertical temperature gradient, while tides play an important role in determining the TL variability of deeper sources (e.g. pile driving) since they cause adjustments of positions of subsurface fronts. The seasonal noise patterns radiated by a large cargo ship are modelled by relating the AIS ship track data and the coupled model, showing a clear influence of the seasonal thermocline and associated bottom fronts on shipping noise distribution. The noise propagates much further (tens of kilometres) in winter than in summer. The predicted shipping noise exposure perceived by grey seals shows strong step changes in the sound level during their descent/ascent through the water column. Since grey seals tend to be benthic foragers, a hypothesis that the step change in sound exposure may have negative impacts on their foraging behaviour is proposed for biological specialists.

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.

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy