Aplicações das séries de Fourier

Abstract

The focus of this project is the introduction to the study of Fourier series. The Fourier series that emerged from the studies of Jean Baptiste Joseph Fourier on the heat equation is an important tool in solving certain types of partial differential equations, such as the heat equation, wave equation, etc. In this work we intend to begin the study of the basic concepts of the Fourier series and, later, to use this tool in the study of some differential equations, such as the heat equation, and finally, to present and demonstrate the Isoperimetric Problem. We will begin by reporting a little of the history of Jean Baptiste Joseph Fourier and his studies on heat, to later study the essential contents for an introduction to the study of Fourier series. The orthogonality of functions will be worked on, in particular the trigonometric functions sine and cosine, and also the properties of periodicity. As motivation for the study of Fourier series, we will follow Fourier’s steps in one of his studies on heat and verify that the finite sum of sines is a solution to the heat problem. Finally we will calculate and graphically sketch some Fourier series and relate them to the convergence theorems and also briefly relate them to the Gibbs phenomenon.

In the final chapters of the work, we will present in detail two applications of Fourier series, the first involving heat conduction problems in a thin bar, with different boundary conditions, and we will solve these problems using Fourier series. Finally, we will present the Isoperimetric Problem, which is essentially a geometric problem, and we will look at this problem from an analytical point of view to use the series in its resolution.