(untitled) (Analysis in Theory and Applications, 2011)
We consider the three dimensional Cauchy problem for the Laplace equation{arrayllu_xx(x,y, z)+u_yy(x,y, z)+u_zz(x,y, z) = 0, & x R, y R, 0 < z ≤ 1, u(x,y,0) = g(x,y), & x R, y R, u_z(x,y,0) = 0, & x R, y R,array.where the data is given at z = 0 and a solution is sought in the region x,y R, 0 < z < 1. The problem is ill-posed, the solution (if it exists) doesn’t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.