DSpace Collection:https://repositorio.ufu.br/handle/123456789/196782024-04-21T09:00:44Z2024-04-21T09:00:44ZO Cálculo Fracionário aplicado à modelos matemáticos clássicoshttps://repositorio.ufu.br/handle/123456789/411512024-02-10T06:18:51Z2023-12-01T00:00:00ZTitle: O Cálculo Fracionário aplicado à modelos matemáticos clássicos
Abstract: This work presents a study on non-integer order calculus, known as Fractional Calculus, applied to classical mathematical models. It was observed that replacing the usual derivative of an ordinary differential equation by a fractional derivative according to Caputo of lower order than the original problem, the solution behavior
of three of the four models studied showed a decrease in the rate of variation. On the other hand, for the last model examined, namely the Malthusian Population Growth model, reducing the order of the problem resulted in a solution with a higher rate of change.2023-12-01T00:00:00ZPoliedros regulares e semirregulareshttps://repositorio.ufu.br/handle/123456789/409812024-01-17T06:17:23Z2023-12-01T00:00:00ZTitle: Poliedros regulares e semirregulares
Abstract: In this Undergraduate Final Project we present a study of the five regular polyhedra (or
Platonic Solids) and the thirteen semi-regular polyhedra (or Archimedean Solids). This study
was divided into two parts: theoretical part, with deductions from formulas and classification of
above polyhedra and; practical part, with the dynamic construction of semi-regular polyhedra
from regular polyhedra using the GeoGebra software.
It is important to emphasize the impossibility of dynamically constructing semi-regular
polyhedra in GeoGebra without some of the formulas related to regular polyhedra that were
deduced in the theoretical part. The formulas deduced were: (i) Euler Relation for convex
polyhedra; (ii) measurements of the central and dihedral angles of a regular polyhedron depending on its genus of faces and its genus of vertices; (iii) radii of spheres inscribed and
circumscribed by a regular polyhedron depending on the genus of faces, the genus of vertices
and the length of the edges; (iv) apothem and radius of the circle circumscribed to the face
of a regular polyhedron depending on its genus of faces and its edge length and; (v) area and
volume of a regular polyhedron as a function of its genus of faces, its genus of vertices, its
number of faces and its edge length.
It is also important to highlight that the constructions of semi-regular polyhedra were made
through three geometric operations on regular polyhedra: (1) simple truncation; (2) composite
truncation and; (3) snubification.
In this work we hope to be able to contribute to the theory of regular and semi-regular
polyhedra, as well as the dynamic geometric constructions of such polyhedra in the GeoGebra
software2023-12-01T00:00:00ZEstudo e aplicação do método multimalha na resolução de sistemas lineareshttps://repositorio.ufu.br/handle/123456789/409442024-01-11T06:18:10Z2023-12-01T00:00:00ZTitle: Estudo e aplicação do método multimalha na resolução de sistemas lineares
Abstract: The search for solutions to linear systems appears in different contexts, such as in the field of optimization, control theory, computational tomography, and the study of fluid dynamics problems. Physical phenomena related to fluid dynamics are often modeled using differential equations, which often lack a known analytical solution. In particular, there is a frequent need to obtain solutions for elliptic partial differential equations, such as the Poisson equation or the Laplace equation. The search for a numerical solution to this type of equation requires solving sparse linear systems of large dimension and not always with well-defined structure, which usually also need to be solved numerically.In this context, the present work focuses on the development and implementation of multigrid methods to investigate their characteristics and evaluate their efficiency in solving systems. This technique is widely used to accelerate the convergence of equation systems by exploiting the advantage that some numerical methods have in effectively smoothing high-frequency errors at different refinement levels. Multigrid methods in a V-cycle composed of 2, 3, or four levels are used to solve sparse linear systems resulting from the discretization of the Laplace and Poisson equations via second-order accurate finite differences. Comparative analyses of runtime and the number of cycles needed were conducted and demonstrate the potential use of the technique as a convergence accelerator for large-scale problems.2023-12-01T00:00:00ZAplicação do Teste da Segunda Derivada para a análise de pontos críticoshttps://repositorio.ufu.br/handle/123456789/400222023-12-22T06:18:06Z2023-11-28T00:00:00ZTitle: Aplicação do Teste da Segunda Derivada para a análise de pontos críticos
Abstract: This work aims to identify the importance of the applications of derivatives in high school. The specific objectives outlined were to: demonstrate the rules of differentiation and emphasize the usefulness of the second derivative. It presents a sequence of concepts and examples to introduce the intuitive ideas of limits and derivatives. Then we study the extrema of functions, Rolle's Theorem, and the Second Derivative Test. At the end of the work, it brings an application in a real problem situation, where it shows an activity to be developed with high school students.2023-11-28T00:00:00Z