Wavelet and Fourier Methods for Solving the Sideways Heat Equation

Lars Eld\'en and Fredrik Berntsson      Teresa Regi\'nska
Department of Mathematics,              Institute of Mathematics
Link\"oping University,                 Polish Academy of Sciences
S-581 83 Link\"oping, Sweden            00-950 Warsaw, Poland

ABSTRACT

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 $0 \leq x
< 1$.

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 approximations or a Fourier--based
approximation. The resulting problem is an initial value problem for
an ordinary differential 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
coefficients. We also solve a problem  from an industrial application,
with actual measured data.