@Article{ATA-28-2,
author = {de, Roberto, Mattos, Linhares, José,  and Prado, Lopes, Ernesto},
title = {Regularization of an Ill-Posed Hyperbolic Problem and the Uniqueness of the Solution by a Wavelet Galerkin Method},
journal = {Analysis in Theory and Applications},
year = {2012},
volume = {28},
number = {2},
pages = {125--134},
abstract = {<p style="text-align: justify;">We consider the problem $K(x)u_{xx} = u_{tt}$, $0 &lt; x &lt; 1$, $t \geq 0$, with the boundary condition $u(0, t) = g(t) \in L^2(R)$ and $u_x(0, t) = 0$, where $K(x)$ is continuous and $0 &lt; \alpha \leq K(x) &lt; +\infty$. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on $g$. Considering the existence of a solution $u(x, \cdot) \in H^2(R)$ and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore, we prove the uniqueness of the solution for this problem.</p>},
issn = {1573-8175},
doi = {https://doi.org/10.3969/j.issn.1672-4070.2012.02.003},
url = {https://global-sci.com/article/74063/regularization-of-an-ill-posed-hyperbolic-problem-and-the-uniqueness-of-the-solution-by-a-wavelet-galerkin-method}
}