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University of Macau / Faculty of Science and Technology / Department of Mathematics

In this thesis we analyse particular differential equations that arise in physical situations. This is achieved with the aid of the computer software package called Mathematica. We first describe the basic features of Mathematica highlighting its capabilities in performing calculations in mathematics. Then we consider a first order Newtonian equation representing the trajectory of a particle around a spherical object. Mathematica is used to solve the Newtonian equation both analytically and numerically. Graphical plots of the trajectories of the planetary bodies Mercury, Earth and Jupiter are presented. We attempt a similar analysis for the corresponding relativistic equation governing the orbits of gravitational objects. Only numerical results are possible in this case. We also perform a perturbative analysis of the relativistic equation and determine the amount of perihelion shift. The second equation considered is the Emden-Fowler equation of order two which arises in many physical problems, including certain inhomogeneous cosmological applications. The analytical features of this equation are investigated using Mathematica and the Lie analysis of differential equations. Different cases of the related autonomous form of the Emden-Fowler equation are investigated and graphically represented. Thereafter, we generate a number of profiles of the energy density and the pressure for a particular solution which demonstrates that a numerical approach for studying inhomogeneity, in cosmological models in general relativity, is feasible. / Thesis (M.Sc.)-University of Natal, Durban, 2001.

Silicon carbide as a representative wide band-gap semiconductor has recently received wide attention due to its excellent physical, thermal and especially electrical properties. It becomes a promising material for electronic and optoelectronic device under high-temperature, high-power and high-frequency and intense radiation conditions. During the Silicon Carbide crystal grown by the physical vapor transport process, the temperature gradients induce thermal stresses which is a major cause of the dislocations multiplication. Although large dimension crystal with low dislocation density is required for satisfying the fast development of electronic and optoelectronic device, high dislocation densities always appear in large dimension crystal. Therefore, reducing dislocation density is one of the primary tasks of process optimization. This dissertation aims at developing a transient finite element model based on the Alexander-Haasen model for computing the dislocation densities in a crystal during its growing process. Different key growth parameters such as temperature gradient, crystal size will be used to investigate their influence on dislocation multiplications. The acceptable and optimal crystal diameter and temperature gradient to produce the lowest dislocation density in SiC crystal can be obtained through a thorough numerical investigation using this developed finite element model. The results reveal that the dislocation density multiplication in SiC crystal are easily affected by the crystal diameter and the temperature gradient. Generally, during the iterative calculation for SiC growth, the dislocation density multiples very rapidly in the early growth phase and then turns to a relatively slow multiplication or no multiplication at all. The results also show that larger size and higher temperature gradient causes the dislocation density enters rapid multiplication phase sooner and the final dislocation density in the crystal is higher. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection

M. Tech. Mathematical Technology. / Aims of this research project is to mathematically analyse froth patterns and build a database of the images at different stages of the fermentation process, so that a decision-making procedure can be developed, which enables a computer to react according to what has been observed. This would allow around-the-clock observation which is not possible with humans. In addition, mechanised decision-making would minimize errors usually associated with human actions. Different mathematical algorithms for image processing will be considered and compared. These algorithms have been designed for different image processing situations. In this dissertation the algorithms will be applied to froth images in particular and will be used to simulate the human eye for decision-making in the fermentation process. The preamble of the study will be to consider algorithms for the detection of edges and then analyse these edges. MATLAB will be used to do the pre-processing of the images and to write and test any new algorithms designed for this project.