TY - JOUR
T1 - Regularization of an Ill-Posed Hyperbolic Problem and the Uniqueness of the Solution by a Wavelet Galerkin Method
AU - José Roberto Linhares de Mattos , 
AU - Prado Lopes , Ernesto
JO - Analysis in Theory and Applications
VL - 2
SP - 125
EP - 134
PY - 2012
DA - 2012/06
SN - 28
DO - http://doi.org/10.3969/j.issn.1672-4070.2012.02.003
UR - https://global-sci.org/intro/article_detail/ata/4549.html
KW - ill-posed problem, meyer wavelet, hyperbolic equation, regularization.
AB - <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>