Um estudo rigoroso da equação do calor e da onda

Abstract

From some basic results of mathematical analysis and metric spaces, this text will build the minimum theory of Fourier analysis to solve problems dealing with the heat equation and wave equation with mathematical rigor. Unfortunately, the Fourier analysis will not be dealt with full generality, since this would need more advanced topics, such as Lebesgue integral. After the minimum theory of Fourier analysis has been established, we will address the heat equation and wave equation in one dimension. After that, we will deal with the Dirichlet problem, which consists in solving the steady-state heat equation in a disc. Unfornately, we will not find the most general contitions to solve those problems. We just will find sufficient conditions for ensure the existence and unicity of the solutions. The main motivation for this study is that usually, in the Physics Major, we just find the solutions for these kind of problems using the separation of variables, disregarding the proper justification of why these solutions satisfy all the conditions for really be the solutions. This is insufficient from the point of view of research in Mathematical Physics.