T. Reginska and L. Elde'n:
Stability and Convergence of a Wavelet-Galerkin 
Method for the Sideways Heat Equation.

Lars Eld\'en                            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.  This is 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 discuss the the stability and convergence properties of
a wavelet-Galerkin method for solving the sideways heat equation. The
wavelets are of Meyer type that have compact support in frequency
space. Previous stability results for this method were suboptimal. 
We show that with additional assumptions concerning the
smoothness of the  solution, and concerning the definition of the
wavelets, we can obtain  almost optimal error estimates.