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Análise da estabilidade do método explícito para discretização de equações diferenciais parabólicas por meio de diferenças finitasAnalysis of the stability of the explicit method for discretization of parabolic differential equations by means of finite differences Resumo Este trabalho tem por objetivo estudar os critérios de estabilidade do método numérico para resolução computacional de equações diferenciais parciais parabólicas. O problema físico utilizado nesse estudo de caso consiste na equação de difusividade térmica em uma barra. O método numérico utiliza aproximações das derivadas de forma explícita por meio de diferenças finitas. Foi aplicado o critério de Neumann para verificar a condição de estabilidade e foram realizados simulações para diferentes passos no eixo temporal. Os resultados das simulações são apresentados em gráficos e tabelas que contém os dados das soluções exatas e aproximadas em determinados pontos. Foi possível observar com os dados numéricos que a condição de estabilidade encontrada pelo critério de Neumann é essencial na prática e que o método explícito aproxima bem da solução com a condição de estabilidade satisfeita. Entretanto, há uma limitação para os tamanhos dos passos a serem tomados. Palavras-chave: Diferenças finitas. Método explícito. Estabilidade. Equações diferenciais parabólicas. AbstractThe objective of this Work is study of criteria of stability of numerical method for computational solution of the partial differential equations parabolic.The physic problem used in this case study consists in thermal difusivity equation in a bar . The numerical method uses aproximation of the derivatives of the explicity form by means of finites diferences. Was aplicated de Neumann method for verify the condition of stability and were realized simulations for differents step of temporal axis. The results of simulations are presenteds in graphics and tables that contains the data of the exact and approximate solutions in points determinated. it was possible to observe whith the numericals data that the stability conditiion found by Neumann criterie is essential in practice and that the explicit method approach well to satisfied condition. However there is a limitation for the step size to be taken.

Nanocrystalline NiFe2O4 particles prepared by chemical co-precipitation method were studied using magnetic measurements, 57Fe Mössbauer spectroscopy, X-ray diffraction, and transmission electron microscopy. Fits to Mössbauer spectra, in the range of 4.2 K – 300 K, were done using spin hamiltonians to describe both the electronic and nuclear interactions, a model of superparamagnetic relaxation of two levels (spin ½) and stochastic theory, a log-normal particle size distribution function as well as a dependency of the magnetic transition temperature and the anisotropy constant on particle diameter. We have used evolutionary strategies to fit the more complex Mössbauer spectra line shapes. The nanoparticles have an average size of 7 nm and exhibit superparamagnetism at room temperature. The saturation magnetization (Ms) at 4.2 K was determined from M vs. 1/H plots by extrapolating the value of magnetizations to infinite fields, to 24.21 emu/g and coercivity to 3.15 kOe. A magnetic anisotropy energy constant (K) 1.9´105 J/m3, at 4.2 K, were calculated from magnetization measurements. The synthesis, characterization, and functionalization of magnetic nanoparticles is a highly active area of current research located at the interface between materials science, biotechnology, and medicine. Superparamagnetic iron oxides nanoparticles have unique physical properties and have emerged as a new class of diagnostic probes for multimodal tracking and as contrast agents for magnetic resonance imaging (MRI).

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In this paper, the evolutionary algorithms approach is applied to the parameterization of a mathematical model describing the Mössbauer spectra of nanogranular (or nanoparticle) magnetic systems. These systems exhibit physical properties very different from bulk specimens being of great interest for material science and its use as biosensors, magneto sensors, data storage, and magnetic fluids. The purpose of this work is to compare the performance between the Differential Evolution and the Evolutionary Strategies algorithms to optimize the model parameters which best fit the experimental Mössbauer spectra of nanoscale magnetic particles. Spectra of two samples (α‐iron foil and NiFe2O4 nanoparticles) were recorded, at room temperature, by a conventional Mössbauer spectrometer using a scintillation detector in transmission geometry with a 57Co/Rh source. Fits to Mössbauer spectra were done using spin hamiltonians to describe both the electronic and nuclear interactions; a model of superparamagnetic relaxation of two levels (spin ½) and stochastic theory; a lognormal particle size distribution function as well as a dependency of the magnetic transition temperature and the anisotropy constant on particle diameter. The evolutionary algorithms have been implemented using Python programming language. For comparison, the two algorithms obey the termination criterion of 6,000 evaluations of the objective function. The results presented show the efficiency of these algorithms in the optimization of the parameters and on the fits of the spectra.