Resumo em Inglês:

ABSTRACT We seek investigate the use of fractional derivatives, both analytically and through simulations. We present some models and perform investigations about them, starting with the classic model and the basic definitions to discuss difficulties in constructing a non-artificial fractional model. Also, we analyze the COVID-19 pandemic using a fractional epidemiological SIR model carefully constructed and present numerical results using MATLAB.

Resumo em Inglês:

ABSTRACT A multiobjective impulsive control scheme is proposed to answer how optimal vaccination campaigns should be implemented, regarding two conflicting targets: making the total number of infecteds small and the vaccination campaign as handy as possible. In this paper, a stochastic SIR model is used to better depict the characteristics of a disease in practical terms, where little influences may lead to sudden and unpredictable changes in the behavior of transmissions. This model is extended to analyze the effects of impulsive vaccinations in two phases: the transient regime control, taking into account the necessity to reduce the number of infected individuals to an acceptable level in a finite time, and the permanent regime control, that will act with fixed vaccinations to avoid another outbreak. A parallel version of NSGA-II is used as the multiobjective optimization machinery, considering both the probability of eradication and the vaccination campaign costs. The final result using the proposed framework shows nondominated policies that can guide public managers to decide which is the best procedure to be taken depending on the present situation.

Resumo em Inglês:

ABSTRACT In this work we solve the nonlinear strong evaporation problem in rarefied gas dynamics. The analysis is based on the BGK model, with three-dimensional velocity, derived from the Boltzmann equation. We present the complete development of a closed form solution for evaluating density, velocity and temperature perturbations and the heat flux of a gas. Numerical results are presented and discussed.

Resumo em Inglês:

ABSTRACT The purpose of this research is to connect fixed point methods with certain third-order boundary value problems in new and interesting ways. Our strategy involves an analysis of the problem under consideration within closed and bounded sets. We develop sufficient conditions under which the associated mappings will be contractive and invariant in these sets, which generates new advances concerning the existence, uniqueness and approximation of solutions.

Resumo em Português:

RESUMO Motivados por aplicações recentes em computação gráfica, este trabalho apresenta um estudo teórico e computacional de sistemas de difusão-reação baseados no Gradient Vector Flow (GVF), com foco no comportamento do GVF em relação às singularidades do campo inicial. O estudo teórico parte de uma análise local, independente de condições de fronteira. Em seguida, supõe-se condição de fronteira no infinito e usa-se análise de Fourier para estabelecer condições suficientes para preservação do ponto singular. Finalmente, supõe-se um domínio compacto, com geometria retangular, e analisa-se a preservação de um ponto singular em relação à condição de fronteira usando um método de solução de equações diferenciais parciais (EDPs) baseado em wavelets de Haar. Desenvolvemos também uma implementação de um método direto para a equação estacionária do GVF baseado em diferenças finitas (DF) para comparar com a solução tradicional do Euler explícito, no que diz respeito a singularidade. É discutida a influência da vorticidade no problema de interesse usando a função de linhas de corrente e equação de Helmholtz. Nos experimentos computacionais, consideramos duas condições de fronteira, dois tipos de singularidades e os três métodos numéricos (Euler explícito, diferenças finitas para a equação estacionária, e wavelets) para verificar os resultados teóricos obtidos.

Resumo em Inglês:

ABSTRACT Motivated by recent computer graphics applications, this work presents a theoretical and computational study of diffusion-reaction systems based on Gradient Vector Flow (GVF), focusing on the behavior of GVF concerning the singularities of the initial vector field. The theoretical study starts from a local analysis, regardless of boundary conditions. Then, the boundary condition at infinity is assumed, and Fourier analysis is used to establish sufficient conditions for preserving the singular point. Finally, a compact domain with rectangular geometry is assumed. The preservation of a singular point concerning the boundary condition is analyzed using a method for solving partial differential equations (PDEs) based on Haar wavelets. We have also developed an implementation of a direct method for the GVF stationary equation based on finite differences (DF) to compare with the traditional explicit Euler solution with respect to singularity. The influence of vorticity on the problem of interest is discussed using the streamlines function and the Helmholtz equation. In the computational experiments, we consider two boundary conditions, two types of singularities, and the three numerical methods (explicit Euler, finite differences for the stationary equation, and wavelets) to verify the theoretical results obtained.

Resumo em Inglês:

ABSTRACT In academia, the study of the movement of water in the ground has been widespread since the last century. The same can be determined using the Richards equation. This is a nonlinear parabolic partial differential equation that requires parameters to generate the results of the problem. Some authors have proposed equations that represent the relationship between volumetric moisture and soil water potential, such as Haverkamp and van Genuchten. As main objective of this work, the inverse modeling was implemented to obtain the Richards equation parameters, applying the Luus-Jaakola method. To verify the mathematical model, a sensitivity analysis was performed, which allowed the observation of the effect that each parameter has on the output data, implying a linear dependence. The results produced proved to be satisfactory for the problem analyzed in our research.

Resumo em Inglês:

ABSTRACT We analyze a seasonal SIR model that assumes a periodic treatment rate. Using the Leray-Schauder degree theory, we prove that model shows periodic solutions. This result shows that sustained oscillations in the incidence of the disease are related to the periodic application of a treatment against the disease. So, we can say that the periodic application of treatment can be considered a seasonal driver of the sustained oscillations.

Resumo em Inglês:

ABSTRACT Treatment with antiviral drugs for human immunodeficiency virus type 1 (HIV-1) infection causes a rapid reduction in plasma viral load. Viral decline occurs in several stages and provides information on important kinetic constants of virus replication in vivo and pharmacodynamic properties. We present a mathematical model that not only considers the intracellular phase of the viral life cycle, defined as the time between the infection of a cell and the production of new viral particles, but we also consider that this parameter together with the virus decay are interactive fuzzy numbers.

Resumo em Inglês:

ABSTRACT A Helicopter rotor undergoes unsteady aerodynamic loads ruled by the aeroelastic coupling between the elastic blades and the dynamic wake induced by rotary wings. Modeling the dynamic interaction between the structural and aerodynamic fields is a key point to understand aeroelastic phenomena associated with rotor stability, flow induced vibration and noise generation, among others. In this study, we address the Generalized Dynamic Wake Model, which describes the inflow velocity field at the rotor disk as a superposition of a finite number of induced flow states. It is a mature model that has been validated based on experimental data and numerically investigated from an eigenvalue problem formulation, whose eigenvalues and eigenvectors provide a deeper insight on the dynamic wake behavior. The paper extends the results presented in the literature to date in order to support physical interpretation of inflow states drawn from the finite-state wake model for flight conditions varying from hover to edgewise flight. The discussion of the wake model mathematical formulation is also oriented towards practical engineering applications to fill a gap in the literature.

Resumo em Inglês:

ABSTRACT The simulation of terrestrial ecosystem processes, using numerical biosphere-atmosphere models that can be coupled to the Atmospheric Models, assist in a better diagnosis and forecast of climate and weather. To be able to represent a particular region, biome or ecosystem, the model parameters need to be fitted for local conditions. This work aims to assess the Luus-Jaakola (LJ) method in the optimization of the parameters in a two-stream radiative transfer model applied to a vegetation canopy. Solar radiation components (incident, S ↓, and reflected, S ↑) were measured above a sugarcane crop in a Tropical region from February 17 to 24, 2006. Among the combinations of internal and external iterations evaluated for Luss-Jaakola method, 60/30 (external/internal) iterations presented more precise albedo (α=S↑/S↓) simulated (r2=0.7386) and, for the accuracy of the simulated α, even though the 60/40 combination had the smallest percentual error (6.40%), the 60/30 combination was 0.03% higher. The precision and accuracy of S ↑ was greater with the parameters obtained by the inverse problem with the combination of 60/30 (external/internal) iterations respectively. In general, the behavior of simulated S ↑ at the top of the canopy was underestimated compared to the observed S ↑, especially in the early morning. For the simulated α at the top of the canopy, the model’s overestimation was observed at the lowest values of albedo. When the largest albedos are observed, only at the beginning of the day the model underestimated the values. As shown by the tests result, the parameters optimized by Luus-Jaakola method have an adequate representation of the observed data.