Graduate Theses & Dissertations

Sinc-Collocation Difference Methods for Solving the Gross-Pitaevskii Equation
The time-dependent Gross-Pitaevskii Equation, describing the movement of parti- cles in quantum mechanics, may not be solved analytically due to its inherent non- linearity. Hence numerical methods are of importance to approximate the solution. This study develops a discrete scheme in time and space to simulate the solution defined in a finite domain by using the Crank-Nicolson difference method and Sinc Collocation Methods (SCM), respectively. In theory and practice, the time discretiz- ing system decays errors in the second-order of accuracy, and SCMs are decaying errors exponentially. A new SCM with a unique boundary treatment is proposed and compared with the original SCM and other similar numerical techniques in time costs and numerical errors. As a result, the new SCM decays errors faster than the original one. Also, to attain the same accuracy, the new SCM interpolates fewer nodes than the original SCM, which saves computational costs. The new SCM is capable of approximating partial differential equations under different boundary con- ditions, which can be extensively applied in fitting theory. Author Keywords: Crank-Nicolson difference method, Gross-Pitaevskii Equation, Sinc-Collocation methods
Modeling drought derivatives in arid regions
We propose a stochastic weather model based on temperature, precipitation, humidity and wind speed for Qatar, as a representative arid region, in order to obtain simulated values for a drought index. As a drought index, the Reconnaissance Drought Index (RDI) is commonly accepted in agriculture and is used to measure drought severity. It can be used to price weather derivatives to help farmers reduce nancial losses from drought. RDI, which is the ratio of precipitation to evapotranspiration, is calculated by considering crop growth stages. The use of dierent crop coecient value depending on the growth stage to calculate evapotranspiration can provide improved values for RDI. Additionally, six calculation methods for evapotranspiration using weather data are investigated to obtain accurate values for RDI. Author Keywords: Evapotranspiration, Markov chains, Mean reversion processes, Reconnaissance Drought Index, Stochastic dierential equations, Stochastic weather models
An Application of the Sinc-Collocation Method in Oceanography
In this thesis, we explore the application of the Sinc-Collocation method to an oceanography model. The model of interest describes a wind-driven current with depth-dependent eddy viscosity and is formulated in two different systems; a complex-velocity system and a real-value coupled system. In general, the Sinc-based methods excel over other traditional numerical methods due to their exponentially decaying errors, rapid convergence and handling problems in the presence of singularities at end-points. In addition, the Sinc-Collocation approach that we utilize exploits first derivative interpolation, whose integration is less sensitive to numerical errors. We present several model problems to demonstrate the accuracy, and stability of the method. We compare the approximate solutions determined by the Sinc-Collocation technique with exact solutions and also with those obtained by the Sinc-Galerkin approach in earlier studies. Our findings indicate that the method we utilized outperforms those used in past studies. Author Keywords: Boundary Value Problems, Eddy Viscosity, Oceanography, Sinc Numerical Methods, Wind-Driven Currents
Stability Properties of Disease Models under Economic Expectations
Comprehending the dynamics of infectious diseases is very important in formulating public health policies to tackling their prevalence. Mathematical epidemiology (ME) has played a very vital role in achieving the above. Nevertheless, classical mathematical epidemiological models do not explicitly model the behavioural responses of individuals in the presence of prevalence of these diseases. Economic epidemiology (EE) as a field has stepped in to fill this gap by integrating economic and mathematical concepts within one framework. This thesis investigated two issues in this area. The methods employed are the standard linear analysis of stability of dynamical systems and numerical simulation. Below are the investigations and the findings of this thesis: Firstly, an investigation into the stability properties of the equilibria of EE models is carried out. We investigated the stability properties of modified EE systems studied by Aadland et al. [6] by introducing a parametric quadratic utility function into the model, thus making it possible to model the maximum number of contacts made by rational individuals to be determined by a parameter. This parameter in particular influences the level of utility of rational individuals. We have shown that if rational individuals have a range of possible contacts to choose from, with the maximum of the number of contacts allowable for these individuals being dependent on a parameter, the variation in this parameter tends to affect the stability properties of the system. We also showed that under the assumption of permanent recovery for disease coupled with individuals observing or not observing their immunity, death and birth rates can affect the stability of the system. These parameters also have effect on the dynamics of the EE SIS system. Secondly, an EE model of syphilis infectivity among &ldquo men who have sex with men &rdquo (MSM) in detention centres is developed in an attempt at looking at the effect of behavioural responses on the disease dynamics among MSM. This was done by explicitly incorporating the interplay of the biology of the disease and the behaviour of the inmates. We investigated the stability properties of the system under rational expectations where we showed that: (1) Behavioural responses to the prevalence of the disease affect the stability of the system. Therefore, public health policies have the tendency of putting the system on indeterminate paths if rational MSM have complete knowledge of the laws governing the motion of the disease states as well as a complete understanding on how others behave in the system when faced with risk-benefit trade-offs. (2) The prevalence of the disease in the long run is influenced by incentives that drive the utility of the MSM inmates. (3) The interplay between the dynamics of the biology of the disease and the behavioural responses of rational MSM tends to put the system at equilibrium quickly as compared to its counterpart (that is when the system is solely dependent on the biology of the disease) when subjected to small perturbation. Author Keywords: economic and mathematical epidemiology models, explosive path, indeterminate-path stability, numerical solution, health gap, saddle-path stability, syphilis,

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