California's almond industry, valued at $2.3 billion per year, depends on the pollinator services of honey bees, although pollination by other insects, mainly solitary wild bees, is being investigated as an alternative because of recent declines in the number of honey bee colonies. Our objective is to model the movements of honey bees and determine the conditions under which they will forage in less favorable areas of a tree and its surroundings when other pollinators are present. We hypothesize that foraging in less favorable areas leads to increased movement between trees and increased cross pollination between varieties which is required for successful nut production. We use the Shigesada-Kawasaki-Teramoto model (1979) which describes the density of two species in a two-dimensional environment of variable favorableness with respect to intrinsic diffusions and intra- and interspecific interactions of species. The model is applied to almond pollination by honey bees and other pollinators with environmental favorableness based on the distribution of flowers in trees. Using the spectral-Galerkin method in a rectangular domain, we numerically approximated the two-dimensional nonlinear parabolic partial differential system arising in the model. When cross-diffusion or interspecific effects of other pollinators was high, honey bees foraged in less favorable areas of the tree. High cross-diffusion also resulted in increased activity in honey bees in terms of accelerations, decelerations, and changes in direction, indicating rapid redistribution of densities to an equilibrium state. Empirical analysis of the number of honey bees and other visitors in two-minute intervals to almond trees shows a negative relationship, indicating cross-diffusion effects in nature with the potential to increase movement to a different tree with a more favorable environment, potentially increasing nut production.

Abstract To improve the application of Point Interpolation Method (PIM) in Element Free Galerkin Method (EFG) is the aim of this study. The trait of EFG is using overlap of influence domain between different nodes to construct discretization nodes¡¦ connection. EFG just uses nodal data, but not element. For constructing shape function, EFG has two types of methods, Fitting and Interpolation. Fitting uses Moving Least Square Method (MLS). MLS-EFG has stable effect on numerical analysis; however, users who use it need to choose more numerical parameters and do more computation. Besides, users can not apply boundary conditions directly when using MLS-EFG. Interpolation method applies nodal coordinates to proceed computation, and it called PIM. Boundary conditions could be used directly and less computation is needs while using PIM. However, the coefficient of interpolation function of sample is singular. This study tries to construct Coordination Point Interpolation Method. It owns advantages of both methods that mentioned above, and extra numerical parameters are not needed. It applies the notion of influence domain of MLS-EFG, then search correlative efficient nodes which are contained in near field of sample. The correlative efficient nodes make up matrix that con cause inverse matrix. In addition, via numerical simulations, it shows that CPIM has excellent convergence and accurate solution, and is better that MLS-EFG.

In this study, the Element-Free Galerkin (EFG) method has been used for the analysis of plane stress problems. A computer program has been developed by using FORTRAN language. The moving least squares (MLS) approximation has been used in generating shape functions. The results obtained by the EFG method have been compared with analytical solution and the numerical results obtained by MSC. Patran/Nastran. The comparisons show that the mesh free method gives more accurate results than the finite element approximation with less computational effort.

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[pt] A parte inicial da dissertação apresenta uma discussão sobre o método de Monte Carlo e sobre geradores de realizações de variáveis aleatórias e vetores aleatórios. São analisados geradores baseados em: congruência linear, Método da Transformada Inversa e algoritmo de Metrópolis-Hastings (método de Monte Carlo com Cadeia de Markov MCMC). Posteriormente, são apresentadas técnicas para a análise de vibrações estocásticas em sistemas mecânicos com grau de liberdade, múltiplos graus de liberdade e contínuos. As propriedades dos sistemas são consideradas determinísticas e as forças são caracterizadas por processos estocásticos. Ao longo das seções do trabalho, é mostrado como características da resposta do sistema em deslocamento (média, autocorrelação e densidade espectral) podem ser obtidas a partir das informações sobre o forçamento. A transformada de Fourier, a função densidade eséctral e a função de transferência são importantes ferramentas utilizadas no estudo. No caso de sistemas com múltiplos graus de liberdade, análise é feita por dois métodos: Resposta ao Impulso e Modos Normais. Nos sistemas contínuos, análise é feita através do Método de Monte Carlo. Para isso, foi desenvolvido um exemplo de um sistema mecânico contínuo submetido a dois carregamentos caracterizados por processos estocásticos. Nesse exemplo, o MCMC foi utilizado para gerar amostras dos forçamentos. Em cada realização, o sistema foi discretizado através do Método dos Elementos Finitos e uma aproximação para solução do problema foi obtida utilizando o Método de Galerkin. Nessa parte da dissertação são apresentados a formulação fraca de equação de movimento da viga e o modelo reduzido. Ao longo de todo o trabalho são mostrados exemplos e resultados obtidos através de rotinas desenvolvidas em MATLAB. / [en] The initial of work presents a discussion about the Monte Carlo Method and about the generation of samples of random variables and ran dom vectors. Generators based on linear congruential, on the Inverse Trans form Method and, on Metropolis-Hastings algorithm (Markov Chain Monte Carlo Method MCMC) are analyzed. Later,techniques for vibrations ana lysis of single-degree of freedom, multi-degree of freedom and continuous stochastic mechanical systems are presented. The system properties are con sidered dertministic and, the forces are characterized by random process. Throughout the work sections, it is show how the system´s displacement response (mean, autocorrrelation and spectral density) can be characterized from the force informations. The Fourier transform, the spectral density function and transfer function are powerful tools used in the study. In the case of multi-degree of freedom systems, the analysis is done by two methods: Impukse Response and Normal modes. For continuous systems, it is made by Monte Carlo method. An example of one continuous mechanical system with two random loads were developed. In this example, MCMC is used to generate samples of the forces. For each realization, the problem is discretized by the Finite Element Method and one approximation of the solution is obtained by the Galerkin Method. In this part of the work, the weak formulation of the motion equation of the vibrating beam and the reduced model are presented. Throughout the work many rountines in MATLAB were developed.

Neste trabalho, apresenta-se o resultado da existência de soluções fracas em domínios não-limitados para as equações de Navier-Stokes, desde que a fronteira satisfaça uma certa condição de regularidade que é necessária para a obtenção de estimativas em domínios não-limitados semelhantes à desigualdade de Poincaré em domínios limitados. Apresenta-se o desenvolvimento detalhado do método de Galerkin para as equações de Navier-Stokes em domínios não-limitados com cálculo explícito de várias constantes e com forças externas não nulas. Apresenta-se dois teoremas fundamentais: um fornecendo condições para existência de soluções do problema estacionário e o outro fornecendo condições para existência de soluções do problema não-estacionário. / In the work it is presented results of existence of weak solutions in unbounded doroains for the Navier-Stokes equations. The roain condition to obtain similar results as those for bounded doroains; for e."'Carople the Poincaré inequality; is a certain condition of regularity at the boundary of the doroain. It is presented the detailed developroent of the Galerkin roethod for the t.he Navier-Stokes equations in unbounded doroains ~vith the explicit calculat ions of many constants and ''rith non null externai forces. It is presented two basic theorern: one presenting condition for the existence of solutions for the stationary problem and the other presenting conditions for existence of solution for the non stationary problem.

Neste trabalho, apresenta-se o resultado da existência de soluções fracas em domínios não-limitados para as equações de Navier-Stokes, desde que a fronteira satisfaça uma certa condição de regularidade que é necessária para a obtenção de estimativas em domínios não-limitados semelhantes à desigualdade de Poincaré em domínios limitados. Apresenta-se o desenvolvimento detalhado do método de Galerkin para as equações de Navier-Stokes em domínios não-limitados com cálculo explícito de várias constantes e com forças externas não nulas. Apresenta-se dois teoremas fundamentais: um fornecendo condições para existência de soluções do problema estacionário e o outro fornecendo condições para existência de soluções do problema não-estacionário. / In the work it is presented results of existence of weak solutions in unbounded doroains for the Navier-Stokes equations. The roain condition to obtain similar results as those for bounded doroains; for e."'Carople the Poincaré inequality; is a certain condition of regularity at the boundary of the doroain. It is presented the detailed developroent of the Galerkin roethod for the t.he Navier-Stokes equations in unbounded doroains ~vith the explicit calculat ions of many constants and ''rith non null externai forces. It is presented two basic theorern: one presenting condition for the existence of solutions for the stationary problem and the other presenting conditions for existence of solution for the non stationary problem.

O método SUPG (\"Streamline Upwind Petrov-Galerkin\") é analisado quanto a sua capacidade de estabilizar oscilações numéricas decorrentes de escoamentos convectivo-difusivos, e de manter a consistência nos resultados. Para esta finalidade, é elaborado um programa computacional como uma implementação algorítmica do método, e simulado o escoamento sobre um cilindro fixo a diferentes números de Reynolds. Ao final, é feita uma revelação sobre a solidez do método. Palavras-chave: escoamento, simulação numérica, método Petrov- Galerkin. / The \"Streamline Upwind Petrov-Galerkin\" method (SUPG) is analyzed with regard to its capability to stabilize numerical oscillations caused by convective-diffusive flows, and to maintain consistency in the results. To this aim, a computational program is elaborated as an algorithmic implementation of the method, and simulated the flow around a fixed cylinder at different Reynolds numbers. At the end, a revelation is made on the method\'s robustness. Keywords: flow, numerical simulation, Petrov-Galerkin method.

Neste trabalho, apresenta-se o resultado da existência de soluções fracas em domínios não-limitados para as equações de Navier-Stokes, desde que a fronteira satisfaça uma certa condição de regularidade que é necessária para a obtenção de estimativas em domínios não-limitados semelhantes à desigualdade de Poincaré em domínios limitados. Apresenta-se o desenvolvimento detalhado do método de Galerkin para as equações de Navier-Stokes em domínios não-limitados com cálculo explícito de várias constantes e com forças externas não nulas. Apresenta-se dois teoremas fundamentais: um fornecendo condições para existência de soluções do problema estacionário e o outro fornecendo condições para existência de soluções do problema não-estacionário. / In the work it is presented results of existence of weak solutions in unbounded doroains for the Navier-Stokes equations. The roain condition to obtain similar results as those for bounded doroains; for e."'Carople the Poincaré inequality; is a certain condition of regularity at the boundary of the doroain. It is presented the detailed developroent of the Galerkin roethod for the t.he Navier-Stokes equations in unbounded doroains ~vith the explicit calculat ions of many constants and ''rith non null externai forces. It is presented two basic theorern: one presenting condition for the existence of solutions for the stationary problem and the other presenting conditions for existence of solution for the non stationary problem.

Efficient numerical methods for option pricing is an active field of research. This project has the goal to examine possible ways to improve an established method of numerical pricing. The method is based on an adaptive finite difference method in price and uses the backwards differentiation formula of order 2, BDF2, in time. The project will focus on improvements to the time integration through implementation of discontinuous Galerkin methods, dG. Empirical convergence and accuracy results are obtained for equidistant dG-methods up to order 3 and performance is compared to BDF2. The dG-methods do not succeed in outperforming the BDF2-method when comparing accuracy to time for computation, but they do match the performance. Possible ways for improvements are suggested.