Numerical Solution of an InverseHeat Conduction

We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using internal measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data is given along the line x = 1, and a solution is sought in the interval 0 x < 1. The problem is ill{posed, in the sense that the solution does not depend continuously of the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded operator. We consider stabilization of the problem using a spectral{based approximation and a Galerkin approximation based on wavelets. The resulting problem is an initial value problem for an ordinary di erential equation in the space variable, which can be solved using standard numerical methods, e.g. a Runge Kutta method. We discuss the numerical implementation of spectral and wavelet methods for solving the sideways heat equation. The theory predicts that the spectral method and a method based on Meyer wavelets will give equally good accuracy. Numerical experiments con rm this. Also, the experiments indicate that a method based on Daubechies wavelets, with compact support, gives comparable results. As test problems we take model equations, with constant and variable coe cients, and also a non{linear equation. In addition, we solve a problem from an industrial application, with actual measured data. Preface I would like to thank my supervisor, Professor Lars Eld en, for guidance and support during this work. Without his valuable insight and encouragement I would not have nished it. Also I would like to thank the National Graduate school for Scienti c computing (NGSSC) for their nancial support. The work presented in this thesis is, to a large extent, based on previous results. In particular I would like to mention the paper by Eld en and Reginska [31], and the technical report by Eld en, Berntsson, and Reginska [14], see also the paper by Berntsson, Eld en, Loyd, and Garcia{Padr on [2], in the proceedings to the 10th International Conference on Numerical Methods for Thermal Problems. The experiment in Section 8.2 were based on measurements conducted at the Department of Mechanical Engineering, Link oping university. I am grateful to Dan Loyd and Ricardo Garcia{Padr on, for performing the experiment and for discussions on real world heat conduction problems.