| Summary: | We consider the problem $K(x)u_{xx}=u_{t}$ , $0<x<1$, $tgeq 0$, 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 $V_{j-1}$.
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