Espaços de Sobolev, problemas de contorno e uma introdução ao método das diferenças finitas

Abstract

This work is divided into two parts. The first part aims to provide a comprehensive study of one-dimensional Sobolev spaces, examining their theoretical foundations, essential properties, and applications. These spaces play a fundamental role in mathematical analysis and constitute important tools for solving problems in areas such as one-dimensional differential equations. In particular, this work emphasizes boundary value problems on intervals subject to Dirichlet, Neumann, and Robin boundary conditions. The second part is devoted to the main objective of this work: the study of the Finite Difference Method, a numerical approach for obtaining approximate solutions to differential equations. The presentation of the method begins with the formulation of a boundary value problem with Dirichlet boundary conditions, from which its fundamental principles are introduced, culminating in the discussion of its applications.