Luiz Bevilacqua

A new analytical formulation of retention effects on particle diffusion processes

The ultimate purpose of this paper is to present a new analytical formulation to simulate diffusion with retention in a reactive medium under stable thermodynamic conditions. The analysis of diffusion with retention in a continuum medium is developed after the solution of an equivalent problem using a discrete approach. The new law may be interpreted as the reduction of all diffusion processes with retention to a unifying phenomenon that can adequately simulate the retention effect namely a circulatory motion. It is remarkable that the governing equation requires a fourth order differential term as suggested by the discrete approach. The relative fraction of diffusion particles β is introduced as a control parameter in the diffusion-retention law as suggested by the discrete approach. This control parameter is essential to avoid retention isolated from the diffusion process. Two matrices referring to material properties are introduced and related to the real phenomenon through the circulation hypothesis. The governing equation may be highly non-linear even if the material properties are constant, but the retention effect is a function of the concentration level, that is, β is a function of the concentration.

On the significance of higher order differential terms in diffusion processes

This paper deals with the analysis of diffusion coupled with temporary retention motivated by the challenge to solve the problem of population spreading. Retention may be associated to colonization of the occupied territory in this case. The discrete approach was selected to deal with this problem due to its relative simplicity and straightforward mathematical treatment. Two types of problems are analyzed namely: symmetric spreading with temporary retention, and propagation with temporary retention. It is clearly shown that higher order differential terms must be included in the governing equations of diffusion and propagation to represent the temporary retention effect. Specifically third and fourth order terms are associated to the retention effect in propagation and diffusion processes respectively. Control parameters regulating the relative influence of the diffusion and the retention terms in the governing equations come up naturally from the analysis. After the appropriated operations the finite difference equations reduce to partial differential equations. The control parameters are kept in the partial differential equations. These parameters are essential in the governing equations to avoid uncontrolled accumulation of particles due to the retention effect. The diffusion-retention problem appearing in several physicochemical problems are governed by the same equations derived here. The current literature refers to several types of diffusion-retention problems, but all solutions assume the classical second order equation as the basic reference. A short analysis of the equilibrium conditions for diffusion-retention problems with a source helps to show the coherence of the theory. In order to explore the potentialities of the discrete approach the problem of asymmetric distribution is also analyzed.

A variational principle for the laplace's operator with application in the torsion of composite rods

Abstract The Ritz method with relaxed coordinate functions is used here to establish a minimizing sequence for an extended functional. The integrand of the functional involves discontinuous functions, which implies that the gradient of the related extremal curve or surface also admits discontinuities. It is discussed a systematic way to form an extended functional allowing the minimizing sequence to approach the exact solution displaying the discontinuities in the gradient of the extremal surface. The extra terms added to the classical functional are shown to be related to the Erdman-Weierstrass corner conditions. Finally the method is applied to the torsion of a composite rod, and the results compared with a solution obtained through the finite elements method.

Modelling and numerical simulation of the velocity field in the Parque Estadual do Cantão (TO), Brazil

The Parque Estadual do Cantao is located at Tocantins (TO), north of Brazil, in a delta formed by the rivers Javaes, Coco and Araguaia. It is a wetland where the seasonal changes drive the biogeochemistry cycles. The knowledge of the water flux dynamics is very important for the understanding of these cycles. This work uses a methodology to model and simulate the velocity field based on Navier-Stokes equations.

Population Dynamics Modeling of Arapaima gigas

Pirarucu (Arapaima gigas) has been of the most important natural fishing resources of the Amazon region. Due to its economic importance, and the necessity to preserve the species hand, field research concerning the habits and behavior of the pirarucu has been increasing for the last 20 years. The aim of this paper is to present a mathematical model for the pirarucu population dynamics considering the species peculiarities, particularly the male parental care over the offspring. The solution of the dynamical systems indicates three possible equilibrium points for the population. The first corresponds to extinction; the third corresponds to a stable population close to the environmental carrying capacity. The second corresponds to an unstable equilibrium located between extinction and full use of the carrying capacity. It is shown that lack of males’ parental care closes the gap between the point corresponding to the unstable equilibrium and the point of stable non-trivial equilibrium. If guarding failure reaches a critical point the two points coincide and the population tends irreversibly to extinction. If some event tends to destabilize the population equilibrium, as for instance inadequate parental care, the model responds in such a way as to restore the trajectory towards the stable equilibrium point avoiding the route to extinction. The parameters introduced to solve the system of equations are partially derived from limited but reliable field data collected at the Mamiraua Sustainable Development Reserve (MSDR) in the Brazilian Amazonian Region.

Inverse Problem of an Anomalous Diffusion Model Employing Lightning Optimization Algorithm

The classical diffusion equation models the behavior of several physical phenomena related to dispersion processes with great success. However, in some cases this approach does not represent the actual physical behavior. In most published works dealing with this situation, the well-known second-order parabolic equation is assumed to be the basic equation of the dispersion process, but the anomalous diffusion effect is modeled with the introduction of fractional derivatives or the imposition of a convenient variation of the diffusion coefficient with time or concentration. Alternatively, Bevilacqua et al. (Process Ann Braz Acad Sci 83:1443–1464, 1977) developed a new analytical formulation for the simulation of the diffusion phenomena with retention. Its purpose is to reduce all retention diffusion processes to a unifying phenomenon that can properly simulate the retention effect. This model may have relevant applications in different areas, such as population dispersion with partial retention of the population to ensure territorial dominance, chemical reactions inducing adsorption processes, and multiphase flow through porous media. In this chapter we are interested in the formulation and solution of the inverse problem involving the anomalous diffusion phenomenon. A two-step solution strategy for the inverse problem is employed because of the correlation between the model parameters, and it is used the Lightning Optimization Algorithm (LOA), a new heuristic that makes an analogy to the phenomena of atmospheric discharges.

O Efeito dos Mosquitos Transgênicos e da Profilaxia na Dinâmica Sazonal de Transmissão da Malária

Neste trabalho e apresentado um modelo matematico que descreve a dinamica de transmissao da malaria com densidade de mosquitos variando sazonalmente; esse modelo e representado por um sistema de equacoes diferenciais ordinarias. Como mecanismos de reducao da incidencia da doenca atuam no modelo dois agentes: (i) a profilaxia, composta por um conjunto de fatores que reduzem o periodo infeccioso dos humanos; (ii) a introducao de mosquitos transgenicos, que sofreram uma alteracao genetica cujo proposito e impedir que o protozoario atinja suas glandulas salivares. O modelo destaca ainda a influencia do aumento da temperatura ambiental, favorecendo a proliferacao de mosquitos e exigindo maior intensidade dos mecanismos de controle.

Inverse Analysis of a New Anomalous Diffusion Model Employing Maximum Likelihood and Bayesian Estimation

The classical diffusion equation models the behavior of several physical phenomena related to dispersion processes quite successfully; however, in some cases, this approach fails to represent the actual physical behavior. In most published works dealing with this situation, the well-known second order parabolic equation is assumed as the basic governing equation of the dispersion process, but the anomalous diffusion effect is modeled with the introduction of fractional derivatives or the imposition of a convenient variation of the diffusion coefficient with time or concentration. Alternatively, Bevilacqua and coauthors developed a new analytical formulation for the simulation of the phenomena of diffusion with retention. Its purpose is to reduce all diffusion processes with retention to a unifying phenomenon that can adequately simulate the retention effect. This model may have relevant applications in different areas, such as population spreading with partial hold up of the population to guarantee territorial domain, chemical reactions inducing adsorption processes, and multiphase flow through porous media. In the new formulation, a discrete approach is first formulated with regard to a control parameter that represents the fraction of particles allowed to diffuse, and the governing equation for the modeling of diffusion with retention in a continuum medium requires a fourth order differential term. In order to apply this new formulation to the modeling of practical problems, the newly introduced parameters need to be accurately determined through an inverse problem analysis. Hence, this chapter provides an overview of the inverse analysis of anomalous diffusion problems as modeled through this new formulation, and a summary is also presented on the inverse problem formulation and related solution through three different approaches: (1) the maximum likelihood estimation, (2) the Bayesian approach through the Maximum a Posteriori objective function, and (3) Markov Chain Monte Carlo methods.

Uma Versão Modificada do Método Evolução Diferencial para Estimativa de Parâmetros do Modelo de Difusão Anômala Assimétrica

No contexto de solucao de problemas inversos, formulados implicitamente, os metodos de otimizacao estocasticos desempenham um papel fundamental para a estimativa de parâmetros. Entre esses metodos, destaca-se o algoritmo Evolucao Diferencial (Differential Evolution – DE). Neste trabalho, e apresentada uma versao modificada deste metodo para estimar parâmetros de um novo modelo para o fenomeno de difusao anomala assimetrica. Os resultados apresentados neste trabalho indicam que com esta modificacao, e possivel diminuir o tempo de CPU atraves da reducao do numero de iteracoes empregando uma reducao do espaco de busca para cada incognita.

Aplicação dos Métodos Evolução Diferencial e Algoritmo de Colisão de Partículas na Estimativa de Parâmetros do Modelo de Difusão Anômala

Neste trabalho e realizada uma abordagem para o problema inverso do modelo de difusao anomala atraves dos metodos estocasticos Particle Collision Algorithm – PCA e Differential Evolution – DE, cujo objetivo e estimar os parâmetros envolvidos no modelo matematico. E apresentada uma aproximacao para a solucao do problema direto empregando o metodo de diferencas finitas, a qual e verificada com uma aproximacao obtida NDSolve Routine do software Mathematica. Uma analise de sensibilidade e realizada para escolha de dados experimentais sinteticos e para um estudo de correlacao entre os parâmetros.