Técnicas de Modelagem Matemática e os Métodos de Runge-Kutta

Abstract

This 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.