ABSTRACT. Advection-diffusion equations are widely used in modeling a diverse range of problems. These mathematical models consist in a partial differential equation or system with initial and boundary conditions, which depend on the phenomena being studied. In the modeling, boundary conditions may be neglected and unnecessarily simplified, or even misunderstood, causing a model not to reflect the reality adequately, making qualitative and/or quantitative analyses more difficult. In this work we derive a general linear flux dependent boundary condition for advection-diffusion problems and show that it generates all possible boundary conditions, according to the outward flux on the boundary. This is done through an integral formulation, analyzing the total mass of the system. We illustrate the exposed cases with applications willing to clarify their meanings. Numerical simulations, by means of the Finite Difference Method, are used in order to exemplify the different boundary conditions' impact, making it possible to quantify the flux along the boundary. With qualitative and quantitative analysis, this work can be useful to researchers and students working on mathematical models with advection-diffusion equations.

Eu agradeço. Por tudo e a todos. Mas mais aos queridos. Ao mestre, com carinho. A João, por tudo. À família. A que eu já tinha quando cheguei: minha mãe, meu pai e meu irmão. E a que a ela somei: Mari, Gustavo, Maíra, Meire, Letícia e Felipe. Aos meus avós porque criaram bem meus pais e por tão mais. À Bruna, por seu bom senso e equilíbrio. Aos amigos que não temem a loucura, ou disfarçam bem: Juliana, Marjory e Márcia. Aos que não menciono. Aos que esqueci. Eu agradeço. Por tudo e a todos. Mas mais aos queridos.

The goal attained by this thesis is the creation and validation of a diffusion coefficient, relative to ecological problems, recovery technique. Eight Chapters constitutes this work and three of them are dedicated to the tools needed for the diffusive related data generation or to the data fit. For the data generation, the numerical solution of a tipically ecological diffusive problem is obtained by combining Finite Elements, Galerkin's Method and Crank-Nicolson. Then, relying on the concepts of probability density function and cumulative distribution function, the Inverse Transform Method is applied. There is a Chapter dedicated to the fitting methods used here to introduce, to a reader who have not had the pleasure of meeting before, an application of a Genetic Algorithm. Such algorithm is used to obtain a non-linear Least Squares three parameter solution. The traditional Linear Regression is also used for fitting another version of the model. The main content, the development of the model and the excellent results compose two more Chapters, the fifth and the seventh that, when gathered to the Introduction, discussion about the bibliography and the Conclusion, closes the work. The kind of data related to ecological problems and the difficulties inherent to it are a main concern and deeply discussed. It is also focus of intense attention how the knowledge on this kind of data is fundamental for the method design. At each step, tool developed or introduced, its quality is attested with the purpose that, by the end of the work, the structure has been built on a solid basis. The method recovers succesfully diffusion coeficients with all the range analyzed, 0.0001 and 1 units of squared space over time; and it is also possible to determine the treatment for the data in such a way that error are directed to a hiper-estimative; that might be provident attitude depending on the problem.