This research is based on a detailed description of model building for multivariate time series models. Under the assumption of stationarity, identification, estimation of the parameters and diagnostic checking for the Vector Auto regressive (p) (VAR(p)), Vector Moving Average (q) (VMA(q)) and Vector Auto regressive Moving Average (VARMA(p, q) ) models are described in detail. With reference to the non-stationary case, the concept of cointegration is explained. Procedures for testing for cointegration, determining the cointegrating rank and estimation of the cointegrated model in the VAR(p) and VARMA(p, q) cases are discussed. The utility of multivariate time series models in the field of economics is discussed and its use is demonstrated by analysing quarterly South African inflation and wage data from April 1996 to December 2008. A review of the literature shows that multivariate time series analysis allows the researcher to: (i) understand phenomenon which occur regularly over a period of time (ii) determine interdependencies between series (iii) establish causal relationships between series and (iv) forecast future variables in a time series based on current and past values of that variable. South African wage and inflation data was analysed using SAS version 9.2. Stationary VAR and VARMA models were run. The model with the best fit was the VAR model as the forecasts were reliable, and the small values of the Portmanteau statistic indicated that the model had a good fit. The VARMA models by contrast, had large values of the Portmanteau statistic as well as unreliable forecasts and thus were found not to fit the data well. There is therefore good evidence to suggest that wage increases occur independently of inflation, and while inflation can be predicted from its past values, it is dependent on wages. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

For a uniquely defined subset of phase space, solutions of non-linear, coupled reaction-diffusion equations may converge to heterogeneous steady states, organic in appearance. Hence, many theoretical models for pattern formation, as in the theory of morphogenesis, include the mechanics of reaction-diffusion equations. The standard method of simulation for such pattern formation models does not facilitate reproducibility of results, or the verification of convergence to a solution of the problem via the method of mesh refinement. In this thesis we explore a new methodology circumventing the aforementioned issues, which is independent of the choice of programming language. While the new method allows more control over solutions, the user is required to make more choices, which may or may not have a determining effect on the nature of resulting patterns. In an attempt to quantify the extent of the possible effects, we study heterogeneous steady states for two well known reaction-diffusion models, the Gierer-Meinhardt model and the Schnakenberg model. / Alexander Graham Bell Canada Graduate Scholarship provides financial support to high calibre scholars who are engaged in master's or doctoral programs in the natural sciences or engineering. / Natural Sciences and Engineering Research Council of Canada

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method. Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations. We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems. To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method. The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions. For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions.

Barrier options are becoming more popular, mainly due to the reduced cost to hold a barrier option when compared to holding a standard call/put options, but exotic options are difficult to price since the payoff functions depend on the whole path of the underlying process, rather than on its value at a specific time instant. It is a path dependent option, which implies that the payoff depends on the path followed by the price of the underlying asset, meaning that barrier options prices are especially sensitive to volatility. For basic exchange traded options, analytical prices, based on the Black-Scholes formula, can be computed. These prices are influenced by supply and demand. There is not always an analytical solution for an exotic option. Hence it is advantageous to have methods that efficiently provide accurate numerical solutions. This study gives a literature overview and compares implementation of some available numerical methods applied to barrier options. The three numerical methods that will be adapted and compared for the pricing of barrier options are: • Binomial Tree Methods • Monte-Carlo Methods • Finite Difference Methods / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2013

Barrier options are becoming more popular, mainly due to the reduced cost to hold a barrier option when compared to holding a standard call/put options, but exotic options are difficult to price since the payoff functions depend on the whole path of the underlying process, rather than on its value at a specific time instant. It is a path dependent option, which implies that the payoff depends on the path followed by the price of the underlying asset, meaning that barrier options prices are especially sensitive to volatility. For basic exchange traded options, analytical prices, based on the Black-Scholes formula, can be computed. These prices are influenced by supply and demand. There is not always an analytical solution for an exotic option. Hence it is advantageous to have methods that efficiently provide accurate numerical solutions. This study gives a literature overview and compares implementation of some available numerical methods applied to barrier options. The three numerical methods that will be adapted and compared for the pricing of barrier options are: • Binomial Tree Methods • Monte-Carlo Methods • Finite Difference Methods / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2013

Stochastic volatility models on option pricing have received much study following the discovery of the non-at implied surface following the crash of the stock markets in 1987. The most widely used stochastic volatility model is introduced by Heston (1993) because of its ability to generate volatility satisfying the market observations, being non-negative and mean-reverting, and also providing a closed-form solution for the European options. However, little research has been done on Heston model used to price early-exercise options. This presumably is largely due to the absence of a closed-form solution and the increase in computational requirement that complicates the required calibration exercise. This thesis examines the performance of the Heston model versus the Black-Scholes model for the American Style equity option of Microsoft and the index option of S&P 100 index. We employ a finite difference method combined with a Projected Successive Over-relaxation method for pricing an American put option under the Black-Scholes model, while an Alternating Direction Implicit method is utilized to decompose a multi-dimensional partial differential equation into several one dimensional steps under the Heston model. For the calibration of the Heston model, we apply a two step procedure where in the first step we apply an indirect inference method to historical stock prices to estimate diffusion parameters under a probability measure and then use a least squares method to estimate the instantaneous volatility and the market risk premium which are used to switch from working under the probability measure to working under the risk-neutral measure. We find that option price is positively related with the value of the mean reverting speed and the long-term variance. It is not sensitive to the market price of risk and it is negatively related with the risk free rate and the volatility of volatility. By comparing the European put option and the American put option under the Heston model, we observe that their implied volatility generally follow similar patterns. However, there are still some interesting observations that can be made from the comparison of the two put options. First, for the out-of-the-money category, the American and European options have rather comparable implied volatilities with the American options' implied volatility being slightly bigger than the European options. While for the in-the-money category, the implied volatility of the European options is notably higher than the American options and its value exceeds the implied volatility of the American options. We also assess the performance of the Heston model by comparing its result with the result from the Black-Scholes model. We observe that overall the Heston model performs better than the Black-Scholes model. In particular, the Heston model has tendency of underpricing the in-the-money option and overpricing the out-of-the-money option. Whereas, the Black-Scholes model is inclined to underprice both the in-the-money option and the out-of-the-money option.b

Large amounts of hydrocarbon reserves are trapped in fractured reservoirs where fluid flux is far more rapid along fractures than through the porous matrix, even though the volume of the pore space may be a hundred times greater than the volume of the fractures. These are considered extremely challenging in terms of accurate recovery prediction because of their complexity and heterogeneity. Conventional reservoir simulators are generally not suited to naturally fractured reservoirs’ production history simulation, especially when production processes are associated with large pressure and temperature changes that lead to large redistribution of effective stresses, causing natural fracture aperture alterations. In this case, all the effective processes, i.e. hydraulic, thermal and geomechanical, should be considered simultaneously to explain and evaluate the behavior of stress-sensitive reservoirs over the production period. This is called thermo-hydro-mechanical (THM) coupling. In this study, a fully coupled thermo-hydro-mechanical approach is developed to simulate the physical behavior of fractures in a plane strain thermo-poroelastic medium. A hybrid numerical method, which implements both the finite difference method (FDM) and the displacement discontinuity method (DDM), is established to study the pressure, temperature, deformation and stress variations of fractures and surrounding rocks during production processes. This method is straightforward and can be implemented in conventional reservoir simulators to update fracture conductivity as it uses the same grid block as the reservoir grids and requires only discretization of fractures. The hybrid model is then verified with couple of analytical solutions for the fracture aperture variation under different conditions. This model is implemented for some examples to present the behavior of fracture network as well as its surrounding rock under thermal injection and production. The results of this work clearly show the importance of rate, aspect ratio (i.e. geometry) and the coupling effects among fracture flow rate and aperture changes arising from coupled stress, pressure and temperature changes. The outcomes of this approach can be used to study the behavior of hydraulic injection for induced fracturing and promoting of shearing such as hydraulic fracturing of shale gas or shale oil reservoirs as well as massive waste disposal in the porous carbonate rocks. Furthermore, implementation of this technique should be able to lead to a better understanding of induced seismicity in injection projects of all kinds, whether it is for waste water disposal, or for the extraction of geothermal energy.

MODELING AND NUMERICAL ANALYSIS OF SINGLE DROPLET DRYING DALMAZ, Nesip M.Sc., Department of Chemical Engineering Supervisor: Prof. Dr. H. &Ouml / nder &Ouml / ZBELGE Co-Supervisor: Asst. Prof. Dr. Yusuf ULUDAg August 2005, 120 pages A new single droplet drying model is developed that can be used as a part of computational modeling of a typical spray drier. It is aimed to describe the drying behavior of a single droplet both in constant and falling rate periods using receding evaporation front approach coupled with the utilization of heat and mass transfer equations. A special attention is addressed to develop two different numerical solution methods, namely the Variable Grid Network (VGN) algorithm for constant rate period and the Variable Time Step (VTS) algorithm for falling rate period, with the requirement of moving boundary analysis. For the assessment of the validity of the model, experimental weight and temperature histories of colloidal silica (SiO2), skimmed milk and sodium sulfate decahydrate (Na2SO4&amp / #8901 / 10H2O) droplets are compared with the model predictions. Further, proper choices of the numerical parameters are sought in order to have successful iteration loops. The model successfully estimated the weight and temperature histories of colloidal silica, dried at air temperatures of 101oC and 178oC, and skimmed milk, dried at air temperatures of 50oC and 90oC, droplets. However, the model failed to predict both the weight and the temperature histories of Na2SO4&amp / #8901 / 10H2O droplets dried at air temperatures of 90oC and 110oC. Using the vapor pressure expression of pure water, which neglects the non-idealities introduced by solid-liquid interactions, in model calculations is addressed to be the main reason of the model resulting poor estimations. However, the developed model gives the flexibility to use a proper vapor pressure expression without much effort for estimation of the drying history of droplets having highly soluble solids with strong solid-liquid interactions. Initial droplet diameters, which were calculated based on the estimations of the critical droplet weights, were predicted in the range of 1.5-2.0 mm, which are in good agreement with the experimental measurements. It is concluded that the study has resulted a new reliable drying model that can be used to predict the drying histories of different materials.

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abstract: In sub transmission systems, many more raptor deaths have been recorded near metal poles rather than wood poles. The metal pole, which is reliable in structure but also grounded, may increase the risk of electrocution when raptors perch on the insulator. This thesis focuses on evaluating the effectiveness of the raptor guard to prevent both debilitating and lethal electrocutions to local wildlife in 69 kV sub transmission systems. First, the two-dimensional (2D) finite difference methods (FDM) were proposed to solve the Poisson and Laplace equations, which describe the electric field. Second, the verification of the FDM algorithm was made based on a parallel-plate capacitor model. Then, the potential and the electric field were simulated by the raptor-insulator model to evaluate the possibility of flashover and leakage current under various conceivable scenarios. Third, several dielectric performance experiments were implemented to gain insight into the physical property of the raptor guard developed by the Salt River Project (SRP) as an example. The proposed initial-tracking-voltage and time-to-track experiments tested the ability of the guard, which is designed to prevent the tracking phenomenon under a contaminated situation such as rain, fog, and snow. A data acquisition also collected the leakage current data for the comparison of maximum raptor tolerance. Furthermore, the puncture voltage of this guard material was performed by the dielectric breakdown voltage experiment in an oil-covered container. With the combination of the model simulation and the experiments in this research, the raptor guard was proven to be practical and beneficial in sub transmission system. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2016