We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.
The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.
The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.