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 - <p style="text-align: justify;">We consider the three dimensional Cauchy problem for the Laplace equation$$\left\{\begin{array}{ll}u_{xx}(x,y, z)+u_{yy}(x,y, z)+u_{zz}(x,y, z) = 0, &nbsp;&amp; x \in R, y \in R, 0 &lt; z \leq 1,\\u(x,y,0) = g(x,y), &amp; x \in R, y \in R,\\u_z(x,y,0) = 0, &nbsp;&amp; &nbsp;x \in R, y \in R,\end{array}\right.$$where the data is given at $z = 0$ and a solution is sought in the region $x,y \in R$, $0 &lt; z &lt; 1$. 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.</p>