01. Universidade Federal Rural de Pernambuco - UFRPE (Sede)
URI permanente desta comunidadehttps://arandu.ufrpe.br/handle/123456789/1
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Item Método de Runge-Kutta de 4ª ordem para a equação de Schrödinger estacionária com energia zero(2021-12-23) Montenegro, João Gabriel Soares; Bastos, Cristiano Costa; http://lattes.cnpq.br/6385190604693576; http://lattes.cnpq.br/3917123866868446The Schrödinger equation has been solved numerically by several Runge-Kutta methods. The study of this equation considering the system energy being zero, among several other applications, allows an analysis of the binding limit state of a particle in a given quantum system. Thus, in the present work we solve the equation in its zero mode, considering an extrinsic approach to confinement in a one-dimensional region, using the 4th order Runge-Kutta method most used for ODE solutions. Initially, we obtained numerically the wavefunctions for a particle confined in a straight line and in circles of different radius, as they are curved with parameterizations by arc length file. Then we study curves from their curvatures, which advises the study of confinement in Archimedean spirals and in logarithmic spirals. Finally, we study confinement in hypothetical curves that do not yet have defined parameterizations. The results obtained made it possible to analyze the regions in the curves with greater tendencies to occur ionization, which could be used as model for the ionization of molecules and nanostructures with geometries similar to those studied.Item Técnicas de Modelagem Matemática e os Métodos de Runge-Kutta(2021-07-23) Silva, Angelo Antunes da Rocha; Didier, Maria Ângela Caldas; Gondim, João Antônio Miranda; http://lattes.cnpq.br/2674397127545655; http://lattes.cnpq.br/9721552594807972; http://lattes.cnpq.br/9069459979748516This work consists in the study of Mathematical Modeling with numerical analysis of the models. In it we present the steps of a modeling process, define and evaluate a mathematical model and also discuss the technique of modeling by fitting curves through the Minimal Squares Method, as well as by differential equations where we approach some models, among them those which describe a populational growth dynamic and epidemiological models. We also present the methods from Taylor Series and Runge-Kutta for the construction of numeric solutions for a initial value problem. As the main contribution we simulated analytical and numerical solutions for four problems of initial value, analysing the error linked to numerical solutions, aiming to answer questions related to the general formula of the Runge-Kutta method of order 2. To calculate error for a certain range we used L2 norm and a closed formula from Newton-Cotes. The purpose here is to offer material in the subject of Mathematical Modeling which can be used by Mathematics Graduation students as another area that utilizes Differential Calculus as a tool. The simulations were coded using Python and the code may be accessed through the link in the beggining of this essay.