Anal. Theory Appl., 27 (2011), pp. 265-277.

Published online: 2011-08

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We consider the three dimensional Cauchy problem for the Laplace equation$$\left\{\begin{array}{ll}u_{xx}(x,y, z)+u_{yy}(x,y, z)+u_{zz}(x,y, z) = 0, & x \in R, y \in R, 0 < z \leq 1,\\u(x,y,0) = g(x,y), & x \in R, y \in R,\\u_z(x,y,0) = 0, & x \in R, y \in R,\end{array}\right.$$where the data is given at $z = 0$ and a solution is sought in the region $x,y \in R$, $0 < z < 1$. 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.1007/s10496-011-0265-6}, url = {http://global-sci.org/intro/article_detail/ata/4599.html} }We consider the three dimensional Cauchy problem for the Laplace equation$$\left\{\begin{array}{ll}u_{xx}(x,y, z)+u_{yy}(x,y, z)+u_{zz}(x,y, z) = 0, & x \in R, y \in R, 0 < z \leq 1,\\u(x,y,0) = g(x,y), & x \in R, y \in R,\\u_z(x,y,0) = 0, & x \in R, y \in R,\end{array}\right.$$where the data is given at $z = 0$ and a solution is sought in the region $x,y \in R$, $0 < z < 1$. 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.

*Analysis in Theory and Applications*.

*27*(3). 265-277. doi:10.1007/s10496-011-0265-6