Volume 27, Issue 3
Uniform Meyer Solution to the Three Dimensional Cauchy Problem for Laplace Equation

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

Published online: 2011-08

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• Abstract

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.

• Keywords

Laplace equation, wavelet solution, uniform convergence.

41A25, 65D15

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@Article{ATA-27-265, author = {}, title = {Uniform Meyer Solution to the Three Dimensional Cauchy Problem for Laplace Equation}, journal = {Analysis in Theory and Applications}, year = {2011}, volume = {27}, number = {3}, pages = {265--277}, abstract = {

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} }
TY - JOUR T1 - Uniform Meyer Solution to the Three Dimensional Cauchy Problem for Laplace Equation JO - Analysis in Theory and Applications VL - 3 SP - 265 EP - 277 PY - 2011 DA - 2011/08 SN - 27 DO - http://doi.org/10.1007/s10496-011-0265-6 UR - https://global-sci.org/intro/article_detail/ata/4599.html KW - Laplace equation, wavelet solution, uniform convergence. AB -

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.

Jinru Wang & Weifang Wang. (1970). Uniform Meyer Solution to the Three Dimensional Cauchy Problem for Laplace Equation. Analysis in Theory and Applications. 27 (3). 265-277. doi:10.1007/s10496-011-0265-6
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