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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

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