Métodos numéricos para resolução de equações diferenciais ordinárias lineares baseados em interpolação por spline

In this work we have elaborated a spline-based method of solution of inicial value problems involving ordinary differential equations, with emphasis on linear equations. The method can be seen as an alternative for the traditional solvers such as Runge-Kutta, and avoids root calculations in the linear time invariant case. The method is then applied on a central problem of control theory, namely, the step response problem for linear EDOs with possibly varying coefficients, where root calculations do not apply. We have implemented an efficient algorithm which uses exclusively matrix-vector operations. The working interval (till the settling time) was determined through a calculation of the least stable mode using a modified power method. Several variants of the method have been compared by simulation. For general linear problems with fine grid, the proposed method compares favorably with the Euler method. In the time invariant case, where the alternative is root calculation, we have indications that the proposed method is competitive for equations of sifficiently high order.

Parâmetro de regularização em problemas inversos: estudo numérico com a transformada de Radon

In general, an inverse problem corresponds to find a value of an element x in a suitable vector space, given a vector y measuring it, in some sense. When we discretize the problem, it usually boils down to solve an equation system f(x) = y, where f : U Rm ! Rn represents the step function in any domain U of the appropriate Rm. As a general rule, we arrive to an ill-posed problem. The resolution of inverse problems has been widely researched along the last decades, because many problems in science and industry consist in determining unknowns that we try to know, by observing its effects under certain indirect measures. Our general subject of this dissertation is the choice of Tykhonov´s regulaziration parameter of a poorly conditioned linear problem, as we are going to discuss on chapter 1 of this dissertation, focusing on the three most popular methods in nowadays literature of the area. Our more specific focus in this dissertation consists in the simulations reported on chapter 2, aiming to compare the performance of the three methods in the recuperation of images measured with the Radon transform, perturbed by the addition of gaussian i.i.d. noise. We choosed a difference operator as regularizer of the problem. The contribution we try to make, in this dissertation, mainly consists on the discussion of numerical simulations we execute, as is exposed in Chapter 2. We understand that the meaning of this dissertation lays much more on the questions which it raises than on saying something definitive about the subject. Partly, for beeing based on numerical experiments with no new mathematical results associated to it, partly for being about numerical experiments made with a single operator. On the other hand, we got some observations which seemed to us interesting on the simulations performed, considered the literature of the area. In special, we highlight observations we resume, at the conclusion of this work, about the different vocations of methods like GCV and L-curve and, also, about the optimal parameters tendency observed in the L-curve method of grouping themselves in a small gap, strongly correlated with the behavior of the generalized singular value decomposition curve of the involved operators, under reasonably broad regularity conditions in the images to be recovered

Unicidade e discretização para problemas de valor inicial

This paper has two objectives: (i) conducting a literature search on the criteria of uniqueness of solution for initial value problems of ordinary differential equations. (ii) a modification of the method of Euler that seems to be able to converge to a solution of the problem, if the solution is not unique

Modelagem e simulação de um sistema de bombeio mecânico em poços direcionais utilizando parâmetros concentrados

This work aims presenting the development of a model and computer simulation of a sucker rod pumping system. This system take into account the well geometry, the flow through the tubing, the dynamic behavior of the rod string and the use of a induction motor model. The rod string were modeled using concentrated parameters, allowing the use of ordinary differential equations systems to simulate it s behavior

Uma reavaliação do pensamento lógico de George Boole à luz da história da matemática

The aim of the present study is to reevaluate the logical thought of the English mathematician George Boole (1815 - 1864). Thus, our research centers on the mathematical analysis of logic in the context of the history of mathematics. In order to do so, we present various biographical considerations about Boole in the light of events that happened in the 19th century and their consequences for mathematical production. We briefly describe Boole's innovations in the areas of differential equations and invariant theory and undertake an analysis of Boole's logic, especially as formulated in the book The Mathematical Analysis of Logic, comparing it not only with the traditional Aristotelian logic, but also with modern symbolic logic. We conclude that Boole, as he intended, expanded logic both in terms of its content and also in terms of its methods and formal elaboration. We further conclude that his purpose was the mathematical modeling of deductive reasoning, which led him to present an innovative formalism for logic and, because the different ways it can be interpreted, a new conception of mathematics

Aplicação de sistemas multi-classificadores no diagnóstico de falhas em motores de indução trifásicos

Equipment maintenance is the major cost factor in industrial plants, it is very important the development of fault predict techniques. Three-phase induction motors are key electrical equipments used in industrial applications mainly because presents low cost and large robustness, however, it isn t protected from other fault types such as shorted winding and broken bars. Several acquisition ways, processing and signal analysis are applied to improve its diagnosis. More efficient techniques use current sensors and its signature analysis. In this dissertation, starting of these sensors, it is to make signal analysis through Park s vector that provides a good visualization capability. Faults data acquisition is an arduous task; in this way, it is developed a methodology for data base construction. Park s transformer is applied into stationary reference for machine modeling of the machine s differential equations solution. Faults detection needs a detailed analysis of variables and its influences that becomes the diagnosis more complex. The tasks of pattern recognition allow that systems are automatically generated, based in patterns and data concepts, in the majority cases undetectable for specialists, helping decision tasks. Classifiers algorithms with diverse learning paradigms: k-Neighborhood, Neural Networks, Decision Trees and Naïves Bayes are used to patterns recognition of machines faults. Multi-classifier systems are used to improve classification errors. It inspected the algorithms homogeneous: Bagging and Boosting and heterogeneous: Vote, Stacking and Stacking C. Results present the effectiveness of constructed model to faults modeling, such as the possibility of using multi-classifiers algorithm on faults classification

Avaliacao de dados de secagem de suspensoes de polpas de frutas em leito de jorro com alimentacao intermitente

The objective of this work was the development and improvement of the mathematical models based on mass and heat balances, representing the drying transient process fruit pulp in spouted bed dryer with intermittent feeding. Mass and energy balance for drying, represented by a system of differential equations, were developed in Fortran language and adapted to the condition of intermittent feeding and mass accumulation. Were used the DASSL routine (Differential Algebraic System Solver) for solving the differential equation system and used a heuristic optimization algorithm in parameter estimation, the Particle Swarm algorithm. From the experimental data food drying, the differential models were used to determine the quantity of water and the drying air temperature at the exit of a spouted bed and accumulated mass of powder in the dryer. The models were validated using the experimental data of drying whose operating conditions, air temperature, flow rate and time intermittency, varied within the limits studied. In reviewing the results predicted, it was found that these models represent the experimental data of the kinetics of production and accumulation of powder and humidity and air temperature at the outlet of the dryer

Modelagem e simulação de um separador bifásico com inversão de fases induzida

The present work has the main goal to study the modeling and simulation of a biphasic separator with induced phase inversion, the MDIF, with the utilization of the finite differences method for the resolution of the partial differencial equations which describe the transport of contaminant s mass fraction inside the equipment s settling chamber. With this aim, was developed the deterministic differential model AMADDA, wich was admensionalizated and then semidiscretizated with the method of lines. The integration of the resultant system of ordinary differential equations was realized by means of a modified algorithm of the Adam-Bashfort- Moulton method, and the sthocastic optimization routine of Basin-Hopping was used in the model s parameter estimation procedure . With the aim to establish a comparative referential for the results obtained with the model AMADDA, were used experimental data presented in previous works of the MDIF s research group. The experimental data and those obtained with the model was assessed regarding its normality by means of the Shapiro-Wilk s test, and validated against the experimental results with the Student s t test and the Kruskal-Wallis s test, depending on the result. The results showed satisfactory performance of the model AMADDA in the evaluation of the MDIF s separation efficiency, being possible to determinate that at 1% significance level the calculated results are equivalent to those determinated experimentally in the reference works

Filtros acústicos em cristais fonônicos

In this work, we have studied the acoustic phonon wave propagation within the periodic and quasiperiodic superlattices of Fibonacci type. These structures are formed by phononic crystals, whose periodicity allows the raise of regions known as stop bands, which prevent the phonon propagation throughout the structure for specific frequency values. This phenomenon allows the construction of acoustic filters with great technological potential. Our theoretical model were based on the method of the transfer matrix, thery acoustics phonons which describes the propagation of the transverse and longitudinal modes within a unit cell, linking them with the precedent cell in the multilayer structure. The transfer matrix is built taking into account the elastic and electromagnetic boundary conditions in the superllatice interfaces, and it is related to the coupled differential equation solutions (elastic and electromagnetic) that describe each model under consideration. We investigated the piezoelectric properties of GaN and AlN the nitride semiconductors, whose properties are important to applications in the semiconductor device industry. The calculations that characterize the piezoelectric system, depend strongly on the cubic (zinc-bend) and hexagonal (wurtzite) crystal symmetries, that are described the elastic and piezoelectric tensors. The investigation of the liquid Hg (mercury), Ga (gallium) and Ar (argon) systems in static conditions also using the classical theory of elasticity. Together with the Euler s equation of fluid mechanics they one solved to the solid/liquid and the liquid/liquid interfaces to obtain and discuss several interesting physical results. In particular, the acoustical filters obtained from these structures are again presented and their features discussed

Um estudo na teoria do movimento brownianno com viscosidade variável

In this work we investigate the stochastic behavior of a large class of systems with variable damping which are described by a time-dependent Lagrangian. Our stochastic approach is based on the Langevin treatment describing the motion of a classical Brownian particle of mass m. Two situations of physical interest are considered. In the first one, we discuss in detail an application of the standard Langevin treatment (white noise) for the variable damping system. In the second one, a more general viewpoint is adopted by assuming a given expression to the so-called collored noise. For both cases, the basic diffententiaql equations are analytically solved and al the quantities physically relevant are explicitly determined. The results depend on an arbitrary q parameter measuring how the behavior of the system departs from the standard brownian particle with constant viscosity. Several types of sthocastic behavior (superdiffusive and subdiffusive) are obteinded when the free pamameter varies continuosly. However, all the results of the conventional Langevin approach with constant damping are recovered in the limit q = 1

Evolução top-hat hidrodinâmica em campos de velocidades peculiares primordiais

We investigate the cosmology of the vacuum energy decaying into cold dark matter according to thermodynamics description of Alcaniz & Lima. We apply this model to analyze the evolution of primordial density perturbations in the matter that gave rise to the first generation of structures bounded by gravity in the Universe, called Population III Objects. The analysis of the dynamics of those systems will involve the calculation of a differential equation system governing the evolution of perturbations to the case of two coupled fluids (dark matter and baryonic matter), modeled with a Top-Hat profile based in the perturbation of the hydrodynamics equations, an efficient analytical tool to study the properties of dark energy models such as the behavior of the linear growth factor and the linear growth index, physical quantities closely related to the fields of peculiar velocities at any time, for different models of dark energy. The properties and the dynamics of current Universe are analyzed through the exact analytical form of the linear growth factor of density fluctuations, taking into account the influence of several physical cooling mechanisms acting on the density fluctuations of the baryonic component of matter during the evolution of the clouds of matter, studied from the primordial hydrogen recombination. This study is naturally extended to more general models of dark energy with constant equation of state parameter in a flat Universe

Função quadrática

In general, the study of quadratic functions is based on an excessive amount formulas, all content is approached without justification. Here is the quadratic function and its properties from problems involving quadratic equations and the technique of completing the square. Based on the definitions we will show that the graph of the quadratic function is the parabola and finished our studies finding that several properties of the function can be read from the simple observation of your chart. Thus, we built the whole matter justifying each step, abandoning the use of decorated formulas and valuing the reasoning

Incorporação do vínculo de suavidade no ajuste de histórico de reservatórios de petróleo

The history match procedure in an oil reservoir is of paramount importance in order to obtain a characterization of the reservoir parameters (statics and dynamics) that implicates in a predict production more perfected. Throughout this process one can find reservoir model parameters which are able to reproduce the behaviour of a real reservoir.Thus, this reservoir model may be used to predict production and can aid the oil file management. During the history match procedure the reservoir model parameters are modified and for every new set of reservoir model parameters found, a fluid flow simulation is performed so that it is possible to evaluate weather or not this new set of parameters reproduces the observations in the actual reservoir. The reservoir is said to be matched when the discrepancies between the model predictions and the observations of the real reservoir are below a certain tolerance. The determination of the model parameters via history matching requires the minimisation of an objective function (difference between the observed and simulated productions according to a chosen norm) in a parameter space populated by many local minima. In other words, more than one set of reservoir model parameters fits the observation. With respect to the non-uniqueness of the solution, the inverse problem associated to history match is ill-posed. In order to reduce this ambiguity, it is necessary to incorporate a priori information and constraints in the model reservoir parameters to be determined. In this dissertation, the regularization of the inverse problem associated to the history match was performed via the introduction of a smoothness constraint in the following parameter: permeability and porosity. This constraint has geological bias of asserting that these two properties smoothly vary in space. In this sense, it is necessary to find the right relative weight of this constrain in the objective function that stabilizes the inversion and yet, introduces minimum bias. A sequential search method called COMPLEX was used to find the reservoir model parameters that best reproduce the observations of a semi-synthetic model. This method does not require the usage of derivatives when searching for the minimum of the objective function. Here, it is shown that the judicious introduction of the smoothness constraint in the objective function formulation reduces the associated ambiguity and introduces minimum bias in the estimates of permeability and porosity of the semi-synthetic reservoir model

Degradação fotocatalítica oxidativa do fenol utilizando carvão obtido da pirólise de diferentes biomassas

The modern industrial progress has been contaminating water with phenolic compounds. These are toxic and carcinogenic substances and it is essential to reduce its concentration in water to a tolerable one, determined by CONAMA, in order to protect the living organisms. In this context, this work focuses on the treatment and characterization of catalysts derived from the bio-coal, by-product of biomass pyrolysis (avelós and wood dust) as well as its evaluation in the phenol photocatalytic degradation reaction. Assays were carried out in a slurry bed reactor, which enables instantaneous measurements of temperature, pH and dissolved oxygen. The experiments were performed in the following operating conditions: temperature of 50 °C, oxygen flow equals to 410 mL min-1 , volume of reagent solution equals to 3.2 L, 400 W UV lamp, at 1 atm pressure, with a 2 hours run. The parameters evaluated were the pH (3.0, 6.9 and 10.7), initial concentration of commercial phenol (250, 500 and 1000 ppm), catalyst concentration (0, 1, 2, and 3 g L-1 ), nature of the catalyst (activated avelós carbon washed with dichloromethane, CAADCM, and CMADCM, activated dust wood carbon washed with dichloromethane). The results of XRF, XRD and BET confirmed the presence of iron and potassium in satisfactory amounts to the CAADCM catalyst and on a reduced amount to CMADCM catalyst, and also the surface area increase of the materials after a chemical and physical activation. The phenol degradation curves indicate that pH has a significant effect on the phenol conversion, showing better results for lowers pH. The optimum concentration of catalyst is observed equals to 1 g L-1 , and the increase of the initial phenol concentration exerts a negative influence in the reaction execution. It was also observed positive effect of the presence of iron and potassium in the catalyst structure: betters conversions were observed for tests conducted with the catalyst CAADCM compared to CMADCM catalyst under the same conditions. The higher conversion was achieved for the test carried out at acid pH (3.0) with an initial concentration of phenol at 250 ppm catalyst in the presence of CAADCM at 1 g L-1 . The liquid samples taken every 15 minutes were analyzed by liquid chromatography identifying and quantifying hydroquinone, p-benzoquinone, catechol and maleic acid. Finally, a reaction mechanism is proposed, cogitating the phenol is transformed into the homogeneous phase and the others react on the catalyst surface. Applying the model of Langmuir-Hinshelwood along with a mass balance it was obtained a system of differential equations that were solved using the Runge-Kutta 4th order method associated with a optimization routine called SWARM (particle swarm) aiming to minimize the least square objective function for obtaining the kinetic and adsorption parameters. Related to the kinetic rate constant, it was obtained a magnitude of 10-3 for the phenol degradation, 10-4 to 10-2 for forming the acids, 10-6 to 10-9 for the mineralization of quinones (hydroquinone, p-benzoquinone and catechol), 10-3 to 10-2 for the mineralization of acids.

Avaliação do desempenho da paligorsquita modificada e do carvão da pirólise do lodo de esgoto no processo de degradação fotocatalítica do fenol

The uncontrolled disposal of wastewaters containing phenolic compounds by the industry has caused irreversible damage to the environment. Because of this, it is now mandatory to develop new methods to treat these effluents before they are disposed of. One of the most promising and low cost approaches is the degradation of phenolic compounds via photocatalysis. This work, in particular, has as the main goal, the customization of a bench scale photoreactor and the preparation of catalysts via utilization of char originated from the fast pyrolysis of sewage sludge. The experiments were carried out at constant temperature (50°C) under oxygen (410, 515, 650 and 750 ml min-1). The reaction took place in the liquid phase (3.4 liters), where the catalyst concentration was 1g L-1 and the initial concentration of phenol was 500 mg L-1 and the reaction time was set to 3 hours. A 400 W lamp was adapted to the reactor. The flow of oxygen was optimized to 650 ml min-1. The pH of the liquid and the nature of the catalyst (acidified and calcined palygorskite, palygorskite impregnated with 3.8% Fe and the pyrolysis char) were investigated. The catalytic materials were characterized by XRD, XRF, and BET. In the process of photocatalytic degradation of phenol, the results showed that the pH has a significant influence on the phenol conversion, with best results for pH equal to 5.5. The phenol conversion ranged from 51.78% for the char sewage sludge to 58.02% (for palygorskite acidified calcined). Liquid samples analyzed by liquid chromatography and the following compounds were identified: hydroquinone, catechol and maleic acid. A mechanism of the reaction was proposed, whereas the phenol is transformed into the homogeneous phase and the others react on the catalyst surface. For the latter, the Langmuir-Hinshelwood model was applied, whose mass balances led to a system of differential equations and these were solved using numerical methods in order to get estimates for the kinetic and adsorption parameters. The model was adjusted satisfactorily to the experimental results. From the proposed mechanism and the operating conditions used in this study, the most favored step, regardless of the catalyst, was the acid group (originated from quinone compounds), being transformed into CO2 and water, whose rate constant k4 presented value of 0.578 mol L-1 min-1 for acidified calcined palygorskite, 0.472 mol L-1 min-1 for Fe2O3/palygorskite and 1.276 mol L-1 min-1 for the sludge to char, the latter being the best catalyst for mineralization of acid to CO2 and water. The quinones were adsorbed to the acidic sites of the calcined palygorskite and Fe2O3/palygorskite whose adsorption constants were similar (~ 4.45 L mol-1) and higher than that of the sewage sludge char (3.77 L mol-1).