Problemas diretos e inversos em processos de difusão anômala

• Autor(a) principal: Silva, Luciano Gonçalves da
• Data de Publicação: 2016
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações da UERJ
• Texto Completo: http://www.bdtd.uerj.br/handle/1/13626
• topic: Anomalous diffusion; Inverse problems; Optimization; Parameters estimation; Difusão - Modelos matemáticos; Problemas inversos (Equações diferenciais); Otimização combinatória; Estimativa de parâmetro; Difusão Anômala; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA

In this thesis are presented solutions for the direct and inverse problem of the anomalous diffusion models proposed by Bevilacqua, Galeão and collaborators in 2011. The finite difference method was applied in the solution of the direct problem in the anomalous diffusion models with symmetrical and asymmetric distribution, while That in the solution of the model of anomalous transport phenomena two particular analytical solutions are employed, one obtained from a simple substitution of variables and another applying the technique of manufactured analytical solution. For the estimation of the parameters involved in the studied models, solutions were obtained for the inverse problem with implicit formulation and, for this, we considered the sensitivity analysis and the experimental design for the determination of the experimental sets to be used in four different stochastic approaches Optimization: Differential Evolution, Particle Collision Algorithm, Lightning Optimization Algorithm and the Bayesian approach developed using the Monte Carlo Method via Markov Chain. These stochastic methods employed are called attention to the good performance of the Lightning Optimization Algorithm, which is a proposal made in this thesis and inspired by the phenomenon of atmospheric discharges. Similar and even better results were obtained when compared to methods already known in the literature (Canonical Differential Evolution and Canonical Particle Collision Algorithm). Due to the correlation between the parameters of the anomalous diffusion models found in the experiment design and in the sensitivity analysis, a two-step solution is used here, which considers the phenomenon to be purely diffusive at a first instant and then the phenomenon Retention or acceleration for anomalous diffusion with symmetrical distribution or the advection phenomenon in the anomalous diffusion model with asymmetric distribution. The strategy used as well as the methods used to estimate the parameters involved in the models were successful in this work.