Fractional derivatives: probability interpretation and frequency response of rational approximations

The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts.

Implementation of fractional-order electromagnetic potential through a genetic algorithm

Several phenomena present in electrical systems motivated the development of comprehensive models based on the theory of fractional calculus (FC). Bearing these ideas in mind, in this work are applied the FC concepts to define, and to evaluate, the electrical potential of fractional order, based in a genetic algorithm optimization scheme. The feasibility and the convergence of the proposed method are evaluated.

Topological entropy of catalytic sets: Hypercycles revisited

The dynamics of catalytic networks have been widely studied over the last decades because of their implications in several fields like prebiotic evolution, virology, neural networks, immunology or ecology. One of the most studied mathematical bodies for catalytic networks was initially formulated in the context of prebiotic evolution, by means of the hypercycle theory. The hypercycle is a set of self-replicating species able to catalyze other replicator species within a cyclic architecture. Hypercyclic organization might arise from a quasispecies as a way to increase the informational containt surpassing the so-called error threshold. The catalytic coupling between replicators makes all the species to behave like a single and coherent evolutionary multimolecular unit. The inherent nonlinearities of catalytic interactions are responsible for the emergence of several types of dynamics, among them, chaos. In this article we begin with a brief review of the hypercycle theory focusing on its evolutionary implications as well as on different dynamics associated to different types of small catalytic networks. Then we study the properties of chaotic hypercycles with error-prone replication with symbolic dynamics theory, characterizing, by means of the theory of topological Markov chains, the topological entropy and the periods of the orbits of unimodal-like iterated maps obtained from the strange attractor. We will focus our study on some key parameters responsible for the structure of the catalytic network: mutation rates, autocatalytic and cross-catalytic interactions.

Efeito de alterações geométricas na resistência de juntas de sobreposição

A utilização de juntas coladas em aplicações industriais tem vindo a aumentar nos últimos anos, em detrimento dos métodos tradicionais de ligação tais como a soldadura, brasagem, ligações aparafusadas e rebitadas. As juntas de sobreposição simples são o tipo de juntas mais frequentemente utilizadas em aplicações industriais, porque são as mais simples de fabricar. No entanto, a aplicação descentrada da carga neste tipo de junta provoca efeitos de flexão que originam o aparecimento de tensões normais na direção da espessura do adesivo (arrancamento), reduzindo assim a resistência da junta colada. De uma maneira geral, existem dois tipos de métodos para reduzir as concentrações de tensões. O primeiro é utilizar alterações no próprio material, otimizando as propriedades do adesivo e do substrato, enquanto o segundo método envolve alterar a geometria da junta, como por exemplo utilizando filetes de adesivo, chanfros nas extremidades dos substratos, aplicar uma geometria ondulada ou dobrar os substratos na zona de sobreposição, ou ainda utilizar rasgos nos substratos ao longo da sobreposição. Neste trabalho é realizado um estudo experimental e numérico por Elementos Finitos de duas alterações efetuadas à geometria de juntas de sobreposição simples, de modo a aumentar a sua resistência comparativamente às juntas sem alteração geométrica. A primeira condição efetuada foi a utilização de rasgos nas extremidades do comprimento de sobreposição e a segunda foi a utilização de rasgos a meio do comprimento de sobreposição. No final do estudo experimental, verificou-se que a resistência da ligação foi significativamente melhorada com algumas das configurações testadas para cada alteração, e foi possível estabelecer em ambos os casos a configuração ótima. Numa fase posterior, procedeu-se à simulação numérica, que incluiu uma análise de tensões e previsão do comportamento das juntas através de modelos de dano coesivo. A análise permitiu obter os modos de rotura, as curvas força-deslocamento e a resistência das juntas. Obteve-se uma concordância bastante boa com os resultados experimentais, o que mostrou a adequabilidade do método de previsão proposto para estimar o comportamento das juntas.

Synchronization in Von Bertalanffy’s models

Many data have been useful to describe the growth of marine mammals, invertebrates and reptiles, seabirds, sea turtles and fishes, using the logistic, the Gom-pertz and von Bertalanffy's growth models. A generalized family of von Bertalanffy's maps, which is proportional to the right hand side of von Bertalanffy's growth equation, is studied and its dynamical approach is proposed. The system complexity is measured using Lyapunov exponents, which depend on two biological parameters: von Bertalanffy's growth rate constant and the asymptotic weight. Applications of synchronization in real world is of current interest. The behavior of birds ocks, schools of fish and other animals is an important phenomenon characterized by synchronized motion of individuals. In this work, we consider networks having in each node a von Bertalanffy's model and we study the synchronization interval of these networks, as a function of those two biological parameters. Numerical simulation are also presented to support our approaches.

Fractional dynamics in the trajectory control of redundant manipulators

Under the pseudoinverse control, robots with kinematical redundancy exhibit an undesirable chaotic joint motion which leads to an erratic behavior. This paper studies the complexity of fractional dynamics of the chaotic response. Fourier and wavelet analysis provides a deeper insight, helpful to know better the lack of repeatability problem of redundant manipulators. This perspective for the study of the chaotic phenomena will permit the development of superior trajectory control algorithms.

Finite Element Analysis (FEA) applied to heat transfer optimization process of pultrusion die systems

The aim of this study is to optimize the heat flow through the pultrusion die assembly system on the manufacturing process of a specific glass-fiber reinforced polymer (GFRP) pultrusion profile. The control of heat flow and its distribution through whole die assembly system is of vital importance in optimizing the actual GFRP pultrusion process. Through mathematical modeling of heating-die process, by means of Finite Element Analysis (FEA) program, an optimum heater selection, die position and temperature control was achieved. The thermal environment within the die was critically modeled relative not only to the applied heat sources, but also to the conductive and convective losses, as well as the thermal contribution arising from the exothermic reaction of resin matrix as it cures or polymerizes from the liquid to solid condition. Numerical simulation was validated with basis on thermographic measurements carried out on key points along the die during pultrusion process.

Spectral solution for the air stripping pollutants removal dynamic model with non linear steady state conditions

This work deals with the numerical simulation of air stripping process for the pre-treatment of groundwater used in human consumption. The model established in steady state presents an exponential solution that is used, together with the Tau Method, to get a spectral approach of the solution of the system of partial differential equations associated to the model in transient state.

The Influence of Damping on Vibration Induced by High-Speed Trains

Passage of high-speed trains may induce high ground and track vibrations, which, besides increasing wheel, rail and track deterioration, may have a negative impact on the vehicle stability and on the passengers comfort. In this paper two distinct analyses are presented. The first one is dedicated to efficient decoupling of rail and soil vibrations by suggesting new interface materials in rail-sleeper fixing system, i.e. in the part where damping efficiency can be directly controlled and tested. The second analysis concerns with an adequate model of soils damping. Proper understanding and correct numerical simulation of this behaviour can help in suggesting soil improvement techniques.

Conceptual analogy for modelling entrapped air action in hydraulic systems

Hydraulic systems are dynamically susceptible in the presence of entrapped air pockets, leading to amplified transient reactions. In order to model the dynamic action of an entrapped air pocket in a confined system, a heuristic mathematical formulation based on a conceptual analogy to a mechanical spring-damper system is proposed. The formulation is based on the polytropic relationship of an ideal gas and includes an additional term, which encompasses the combined damping effects associated with the thermodynamic deviations from the theoretical transformation, as well as those arising from the transient vorticity developed in both fluid domains (air and water). These effects represent the key factors that account for flow energy dissipation and pressure damping. Model validation was completed via numerical simulation of experimental measurements.

Mathematical model for HIV dynamics in HIV-specific helper cells

In this paper we study a delay mathematical model for the dynamics of HIV in HIV-specific CD4 + T helper cells. We modify the model presented by Roy and Wodarz in 2012, where the HIV dynamics is studied, considering a single CD4 + T cell population. Non-specific helper cells are included as alternative target cell population, to account for macrophages and dendritic cells. In this paper, we include two types of delay: (1) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and; (2) virus production period for new virions to be produced within and released from the infected cells. We compute the reproduction number of the model, R0, and the local stability of the disease free equilibrium and of the endemic equilibrium. We find that for values of R0<1, the model approaches asymptotically the disease free equilibrium. For values of R0>1, the model approximates asymptotically the endemic equilibrium. We observe numerically the phenomenon of backward bifurcation for values of R0⪅1. This statement will be proved in future work. We also vary the values of the latent period and the production period of infected cells and free virus. We conclude that increasing these values translates in a decrease of the reproduction number. Thus, a good strategy to control the HIV virus should focus on drugs to prolong the latent period and/or slow down the virus production. These results suggest that the model is mathematically and epidemiologically well-posed.

Retificador trifásico controlado com fator de potência quase unitário

Dissertação de Mestrado para obtenção do grau de Mestre em Engenharia Eletrotécnica Ramo Automação e Eletrónica Industrial

Retificador monofásico com fator de potência quase unitário

Dissertação para a obtenção do grau de Mestre em Engenharia Eletrotécnica Ramo de Automação e Eletrónica Industrial

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.

Visible range plasmonic effect produced by aluminium nanoparticles embedded in amorphous silicon

We present results, obtained by means of an analytic study and a numerical simulation, about the resonant condition necessary to produce a Localized Surface Plasmonic Resonance (LSPR) effect at the surface of metal nanospheres embedded in an amorphous silicon matrix. The study is based on a Lorentz dispersive model for a-Si:H permittivity and a Drude model for the metals. Considering the absorption spectra of a-Si:H, the best choice for the metal nanoparticles appears to be aluminium, indium or magnesium. No difference has been observed when considering a-SiC:H. Finite-difference time-domain (FDTD) simulation of an Al nanosphere embedded into an amorphous silicon matrix shows an increased scattering radius and the presence of LSPR induced by the metal/semiconductor interaction under green light (560 nm) illumination. Further results include the effect of the nanoparticles shape (nano-ellipsoids) in controlling the wavelength suitable to produce LSPR. It has been shown that is possible to produce LSPR in the red part of the visible spectrum (the most critical for a-Si:H solar cells applications in terms of light absorption enhancement) with aluminium nano-ellipsoids. As an additional results we may conclude that the double Lorentz-Lorenz model for the optical functions of a-Si:H is numerically stable in 3D simulations and can be used safely in the FDTD algorithm. A further simulation study is directed to determine an optimal spatial distribution of Al nanoparticles, with variable shapes, capable to enhance light absorption in the red part of the visible spectrum, exploiting light trapping and plasmonic effects. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.