Numerical Treatment of a Data Completion Problem in Heat Conduction Modelling

This work deals with a question in the mathematical modelling for the temperature evolution in a bar, for a long time linked as an inverse problem. The one-dimensional model is the parabolic partial differential equation u(t) = alpha u(xx), known as the heat diffusion equation. The classic direct problem (DP) involves this equation coupled to a set of constraints: initial and boundary conditions, in such a way as to guarantee existence of a unique solution. The data completion (DC) problem hereby considered may be described as follows: the temperature at one of the bar extreme points is unknown but there is a fixed interior point where it may be measured, for all time. Finite difference algorithms (FDA) were tested to approximate the solution for such a problem. The important point to be emphasized is that FDA may show up distinct performances when applied to either DP or DC, which is due to the way the discrete variables follow up the mesh steps - advancing in time, for the first case, on the space direction, for the other.