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

Jinru Wang & Weifang Wang

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

Published online: 2011-08

Export citation
  • 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.

  • AMS Subject Headings

41A25, 65D15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{ATA-27-265, author = {Jinru Wang and Weifang Wang}, 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 AU - Jinru Wang & Weifang Wang 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 and Weifang Wang. (2011). 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
Copy to clipboard
The citation has been copied to your clipboard