Equações Diferenciais Parciais: uma Solução em Série para a Equação do Calor

Abstract

The Undergraduated Thesis has the theme Partial Differential Equations (PDE), more especifically the heat Equation, addressing concepts related to the Fourier’ Serie, and presentes the search problem: using a technique resolution of partial differential equations – and by means a study of the Fourier’ Serie and the resumption of concepts of differential and integral calculus –, is it possible to get a solution of a problem related to the spread of heat in a bar section straight uniform and homogeneous material? Trying to solve this question, the main aim was to understand a technique resolution partial differential equations, aiming at the solution related to the Equation Heat, following concepts of differential and integral calculus, with emphasis in the Fourier’ Serie. This, because the differential equations are quite used to solve problems about mathematical modeling – mainly in the chemical and physical areas. In solving this problem became necessary recover, enlarge and understand concepts of Calculus, Algebra, Ordinary Differential Equations (ODE) and series, divided into three steps that take into consideration: the transformation of a PDE in two ODE by the method of separation of variables to find a constant of separation and then apply the contour conditions, checking three posibilities to this, in search for a solution nontrivial; the use of the problem of regular Sturm-Liouville, that contributes in determining of the self values and self functions. From there, studies was maked that meet the inicial condition in a serie, being posible to check the frequency and the orthogonal of functions, as wel as determine solutions applying the principle of superposition and the Fourier’ Series due to obtain a serie of sines and, consequently, became necessary the use of the convergence theorem, getting so, a solution in serie to the Equation Heat.