Wavelet and Fourier Methods for Solving the Sideways Heat Equation

We consider an inverse heat conduction problem, the Sideways Heat Equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 x < 1. The problem is ill{posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet{ based approximationsor a Fourier{based approximation. The resulting problem is an initial value problem for an ordinary diierential equation, which can be solved by standard numerical methods, e.g. a Runge{ Kutta method. We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations, with constant and variable coeecients. We also solve a problem from an industrial application, with actual measured data.