Solving  the Sideways Heat Equation by 
a Wavelet-Galerkin Method

Teresa Regi\'nska
Institute of Mathematics,
Polish Academy of Sciences,
00-950 Warsaw, Poland

Lars Eld\'en
Department of Mathematics, Link\"oping University,
 S-581 83 Link\"oping, Sweden


ABSTRACT
We consider a Cauchy problem for the heat equation in the quarter
plane, where data are given at $x=1$ and a solution is sought in the
interval $0 < x < 1$. This {\sl sideways heat equation} 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.
The problem is ill-posed, in the sense that the solution (if it
exists) does not depend continuously on the data.  Meyer wavelets have
the property that their Fourier transform has compact
support. Therefore, by expanding the data and the solution in a basis
of Meyer wavelets, high frequency components can be filtered away. We
show that using a wavelet-Galerkin approach, we restore continuous
dependence on the data, and we give a recipe for choosing the coarse
level resolution in the wavelet representation, depending on the noise
level of the data.  Furthermore, we solve the sideways problem
numerically in the coarse level representation, as an ordinary
differential equation in the space variable, where the time derivative
is replaced by its wavelet representation. Numerical examples are
given.