Electronic Journal of Differential Equations,
Conf. 10 (2002), pp. 211-225.

Title: A wavelet Galerkin method applied to partial differential
  equations with variable coefficients

Authors: Jose Roberto Linhares de Mattos (Rural Federal Univ. of Rio de Janeiro, Brazil)
   Ernesto Prado Lopes (Federal University of Rio de Janeiro, Brazil)
 
Abstract: 
 We consider the problem $K(x)u_{xx}=u_{t}$ , $0<x<1$, $t\geq 0$,
 where $K(x)$ is bounded below by a positive constant.
 The solution on the boundary $x=0$ is a known function $g$
 and $u_{x}(0,t)=0$. This is an ill-posed problem in the sense
 that a small disturbance on the boundary specification $g$,
 can produce a big alteration on its solution, if it exists.
 We consider the existence of a solution $u(x,\cdot)\in L^{2}(R)$
 and we use a wavelet Galerkin method with the Meyer multi-resolution analysis,
 to filter away the high-frequencies and to obtain well-posed approximating
 problems in the scaling spaces $V_{j}$. We also derive an estimate for
 the difference between the exact solution of the problem and the orthogonal
 projection, onto $V_{j}$, of the solution of the approximating problem
 defined in $V_{j-1}$.

Published February 28, 2003.
Math Subject Classifications: 65T60.
Key Words: Wavelet; multi-resolution analysis.