Métodos numéricos para equações diferencias ordinárias

Pós-graduação em Matemática Universitária - IGCE

Modelagem Matemática e Análise de Modelos Matemáticos na Educação Matemática

In this paper we present two studies, the first one completed and the second one in development, which are based in teaching approaches that propose the qualitative study of mathematical models as a strategy for the teaching and learning of mathematical concepts. These teaching approaches focus on subjects from Higher Education such as Introduction to Ordinary Differential Equations and Topics of Differential and Integral Calculus. We denominate this common aspect of the teaching approaches as Model Analysis and in a preliminary level we relate it with Mathematical Modeling. Furthermore, we discuss some questions related with the choice of the theme and the role of Digital Technologies when Model Analysis is applied.

Equações diferenciais ordinárias não suaves autônomas e não autônomas

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Equações diferenciais ordinárias não suaves autônomas e não autônomas

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

On estimating the basic reproduction number in distinct stages of a contagious disease spreading

In epidemiology, the basic reproduction number R-0 is usually defined as the average number of new infections caused by a single infective individual introduced into a completely susceptible population. According to this definition. R-0 is related to the initial stage of the spreading of a contagious disease. However, from epidemiological models based on ordinary differential equations (ODE), R-0 is commonly derived from a linear stability analysis and interpreted as a bifurcation parameter: typically, when R-0 >1, the contagious disease tends to persist in the population because the endemic stationary solution is asymptotically stable: when R-0 <1, the corresponding pathogen tends to naturally disappear because the disease-free stationary solution is asymptotically stable. Here we intend to answer the following question: Do these two different approaches for calculating R-0 give the same numerical values? In other words, is the number of secondary infections caused by a unique sick individual equal to the threshold obtained from stability analysis of steady states of ODE? For finding the answer, we use a susceptibleinfective-recovered (SIR) model described in terms of ODE and also in terms of a probabilistic cellular automaton (PCA), where each individual (corresponding to a cell of the PCA lattice) is connected to others by a random network favoring local contacts. The values of R-0 obtained from both approaches are compared, showing good agreement. (C) 2012 Elsevier B.V. All rights reserved.

Barriers to front propagation in ordered and disordered vortex flows

We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.

Effect of mesh distortion on the accuracy of high order vorticity-velocity CFD approaches

The numerical solution of the incompressible Navier-Stokes equations offers an alternative to experimental analysis of fluid-structure interaction (FSI). We would save a lot of time and effort and help cut back on costs, if we are able to accurately model systems by these numerical solutions. These advantages are even more obvious when considering huge structures like bridges, high rise buildings or even wind turbine blades with diameters as large as 200 meters. The modeling of such processes, however, involves complex multiphysics problems along with complex geometries. This thesis focuses on a novel vorticity-velocity formulation called the Kinematic Laplacian Equation (KLE) to solve the incompressible Navier-stokes equations for such FSI problems. This scheme allows for the implementation of robust adaptive ordinary differential equations (ODE) time integration schemes, allowing us to tackle each problem as a separate module. The current algortihm for the KLE uses an unstructured quadrilateral mesh, formed by dividing each triangle of an unstructured triangular mesh into three quadrilaterals for spatial discretization. This research deals with determining a suitable measure of mesh quality based on the physics of the problems being tackled. This is followed by exploring methods to improve the quality of quadrilateral elements obtained from the triangles and thereby improving the overall mesh quality. A series of numerical experiments were designed and conducted for this purpose and the results obtained were tested on different geometries with varying degrees of mesh density.

Modeling and Vibration Suppression for Telescopic Systems of Structural Members with Clearance

Telescopic systems of structural members with clearance are found in many applications, e.g., mobile cranes, rack feeders, fork lifters, stacker cranes (see Figure 1). Operating these machines, undesirable vibrations may reduce the performance and increase safety problems. Therefore, this contribution has the aim to reduce these harmful vibrations. For a better understanding, the dynamic behaviour of these constructions is analysed. The main interest is the overlapping area of each two sections of the above described systems (see markings in Figure 1) which is investigated by measurements and by computations. A test rig is constructed to determine the dynamic behaviour by measuring fundamental vibrations and higher frequent oscillations, damping coefficients, special appearances and more. For an appropriate physical model, the governing boundary value problem is derived by applying Hamilton’s principle and a classical discretisation procedure is used to generate a coupled system of nonlinear ordinary differential equations as the corresponding truncated mathematical model. On the basis of this model, a controller concept for preventing harmful vibrations is developed.

MODELLING β2AR REGULATION

The β2 adrenergic receptor (β2AR) regulates smooth muscle relaxation in the vasculature and airways. Long- and Short-acting β-agonists (LABAs/SABAs) are widely used in treatment of chronic obstructive pulmonary disorder (COPD) and asthma. Despite their widespread clinical use we do not understand well the dominant β2AR regulatory pathways that are stimulated during therapy and bring about tachyphylaxis, which is the loss of drug effects. Thus, an understanding of how the β2AR responds to various β-agonists is crucial to their rational use. Towards that end we have developed deterministic models that explore the mechanism of drug- induced β2AR regulation. These mathematical models can be classified into three classes; (i) Six quantitative models of SABA-induced G protein coupled receptor kinase (GRK)-mediated β2AR regulation; (ii) Three phenomenological models of salmeterol (a LABA)-induced GRK-mediated β2AR regulation; and (iii) One semi-quantitative, unified model of SABA-induced GRK-, protein kinase A (PKA)-, and phosphodiesterase (PDE)-mediated regulation of β2AR signalling. The various models were constrained with all or some of the following experimental data; (i) GRK-mediated β2AR phosphorylation in response to various LABAs/SABAs; (ii) dephosphorylation of the GRK site on the β2AR; (iii) β2AR internalisation; (iv) β2AR recycling; (v) β2AR desensitisation; (vi) β2AR resensitisation; (vii) PKA-mediated β2AR phosphorylation in response to a SABA; and (viii) LABA/SABA induced cAMP profile ± PDE inhibitors. The models of GRK-mediated β2AR regulation show that plasma membrane dephosphorylation and recycling of the phosphorylated β2AR are required to reconcile with the measured dephosphorylation kinetics. We further used a consensus model to predict the consequences of rapid pulsatile agonist stimulation and found that although resensitisation was rapid, the β2AR system retained the memory of prior stimuli and desensitised much more rapidly and strongly in response to subsequent stimuli. This could explain tachyphylaxis of SABAs over repeated use in rescue therapy of asthma patients. The LABA models show that the long action of salmeterol can be explained due to decreased stability of the arrestin/β2AR/salmeterol complex. This could explain long action of β-agonists used in maintenance therapy of asthma patients. Our consensus model of PKA/PDE/GRK-mediated β2AR regulation is being used to identify the dominant β2AR desensitisation pathways under different therapeutic regimens in human airway cells. In summary our models represent a significant advance towards understanding agonist-specific β2AR regulation that will aid in a more rational use of the β2AR agonists in the treatment of asthma.

Demographic model of the Swiss cattle population for the years 2009-2011 stratified by gender, age and production type

Demographic composition and dynamics of animal and human populations are important determinants for the transmission dynamics of infectious disease and for the effect of infectious disease or environmental disasters on productivity. In many circumstances, demographic data are not available or of poor quality. Since 1999 Switzerland has been recording cattle movements, births, deaths and slaughter in an animal movement database (AMD). The data present in the AMD offers the opportunity for analysing and understanding the dynamic of the Swiss cattle population. A dynamic population model can serve as a building block for future disease transmission models and help policy makers in developing strategies regarding animal health, animal welfare, livestock management and productivity. The Swiss cattle population was therefore modelled using a system of ordinary differential equations. The model was stratified by production type (dairy or beef), age and gender (male and female calves: 0-1 year, heifers and young bulls: 1-2 years, cows and bulls: older than 2 years). The simulation of the Swiss cattle population reflects the observed pattern accurately. Parameters were optimized on the basis of the goodness-of-fit (using the Powell algorithm). The fitted rates were compared with calculated rates from the AMD and differed only marginally. This gives confidence in the fitted rates of parameters that are not directly deductible from the AMD (e.g. the proportion of calves that are moved from the dairy system to fattening plants).

The Devil’s Calculus: Mathematical Models of Civil War

In spite of the movement to turn political science into a real science, various mathematical methods that are now the staples of physics, biology, and even economics are thoroughly uncommon in political science, especially the study of civil war. This study seeks to apply such methods - specifically, ordinary differential equations (ODEs) - to model civil war based on what one might dub the capabilities school of thought, which roughly states that civil wars end only when one side’s ability to make war falls far enough to make peace truly attractive. I construct several different ODE-based models and then test them all to see which best predicts the instantaneous capabilities of both sides of the Sri Lankan civil war in the period from 1990 to 1994 given parameters and initial conditions. The model that the tests declare most accurate gives very accurate predictions of state military capabilities and reasonable short term predictions of cumulative deaths. Analysis of the model reveals the scale of the importance of rebel finances to the sustainability of insurgency, most notably that the number of troops required to put down the Tamil Tigers is reduced by nearly a full order of magnitude when Tiger foreign funding is stopped. The study thus demonstrates that accurate foresight may come of relatively simple dynamical models, and implies the great potential of advanced and currently unconventional non-statistical mathematical methods in political science.

The internal structure of hydrogen-air difussion flames

The Internal Structure of Hydrogen-Air Diffusion Flames. Tho purpose of this paper is to study finite rate chemistry effects in diffusion controlled hydrogenair flames undor conditions appearing in some cases in a supersonic combustor. Since for large reaction rates the flame is close to chemical equilibrium, the reaction takes place in a very thin region, so thata "singular perturbation "treatment" of the problem seems appropriate. It has been shown previously that, within the inner or reaction zone, convection effects may be neglocted, the temperature is constant across the flame, and tho mass fraction distributions are given by ordinary differential equations, whore tho only independent variable involved is tho coordinate normal to the flame surface. Tho solution of the outer problom, which is a pure mixing problem with the additional condition that fuol and oxidizer do not coexist in any zone, provides t h e following information: tho flame position, rates of fuel consumption, temperature, concentrators of species, fluid velocity outside of tho flame, and the boundary conditions required to solve the "inner problem." The main contribution of this paper consists in the introduction of a fairly complicated chemical kinetic scheme representing hydrogen-oxygen reaction. The nonlinear equations expressing the conservation of chemical species are approximately integrated by means of an integral method. It has boen found that, in the case considered of a near-equilibrium diffusion flame, tho role played by the dissociation-recombination reactions is purely marginal, and that somo of the second order "shuffling" reactions are close to equilibrium. The method shown here may be applied to compute the distanco from the injector corresponding to a given separation from equilibrium, say ten to twenty percent. For the casos whore this length is a small fraction of the combustion zone length, the equilibrium treatment describes properly tho flame behavior.

Adams methods: implementation as predictor-corrector methods

In this paper, Adams explicit and implicit formulas are obtained in a simple way and a relationship between them is established, allowing for their joint implementation as predictor-corrector methods. It is shown the purposefulness, from a didactic point of view, of Excel spreadsheets for calculations and for the orderly presentation of results in the application of Adams methods to solving initial value problems in ordinary differential equations.

Low-dimensional modeling of streaks in a wedge flow boundary layer

This paper is concerned with the low dimensional structure of optimal streaks in a wedge flow boundary layer, which have been recently shown to consist of a unique (up to a constant factor) three-dimensional streamwise evolving mode, known as the most unstable streaky mode. Optimal streaks exhibit a still unexplored/unexploited approximate self-similarity (not associated with the boundary layer self-similarity), namely the streamwise velocity re-scaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate; the remaining two velocity components instead do not satisfy this property. The approximate self-similar behavior is analyzed here and exploited to further simplify the description of optimal streaks. In particular, it is shown that streaks can be approximately described in terms of the streamwise evolution of the scalar amplitudes of just three one-dimensional modes, providing the wall normal profiles of the streamwise velocity and two combinations of the cross flow velocity components; the scalar amplitudes obey a singular system of three ordinary differential equations (involving only two degrees of freedom), which approximates well the streamwise evolution of the general streaks.

Mathematical and Numerical Modeling in Maritime Geomechanics

A theoretical and numerical framework to model the foundation of marine offshore structures is presented. The theoretical model is composed by a system of partial differential equations describing coupling between seabed solid skeleton and pore fluids (water, air, oil,…) combined with a system of ordinary differential equations describing the specific constitutive relation of the seabed soil skeleton. Once the theoretical model is described, the finite element numerical procedure to achieve an approximate solution of the governing equations is outlined. In order to validate the proposed theoretical and numerical framework the seaward tilt mechanism induced by the action of breaking waves over a vertical breakwater is numerically reproduced. The results numerically attained are in agreement with the main conclusions drawn from the literature associated with this failure mechanism