Resolució numèrica d’equacions en derivades parcials parabòliques

Abstract

A partial differential equation (PDE) is an equation that contains an unknown function with more than one variable and some of its partial derivatives. We can classify PDE’s in linears or nonlinears and by the order of its derivatives as well. Within these classifications stand out linear PDE’s of second order, because they can modelling a lot of physical phenomena. We find three groups inside this set: • Elliptical equations, which appear in heat transmission under stationary conditions problems, particle difussion or the vibration in a membrane. • Parabolic equations, which appear in the same kind of problems than before but with one exception, they change over time now. • Hyperbolic equations, which appear in problems about mass transport in fluids, wave phenomena, among others. In this work, we will study several types of boundary value problems (BVP) of a parabolic equation, the heat equation. BVP consists in finding a function $f\in C^{2}$ that satisfies the conditions of the heat equation and the conditions imposed on the unknown function (or its derivatives) in the boundary of the region we are working on. Frequently, these problems cannot be solved analytically. One of the most used methods these days is the finite element method. This method was developed from the 40’s. The aim of the finite element method is to approximate a weak solution for a BVP from a mount of referency nodes which are located on the region of the problem. To perform this whole process, we will start describing some functional analysis tools that we will need from now on in this work, straightaway we will see the numerical methods that we will use and we will conclude with an example resolution.