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Volume 28, Issue 2
Regularization of an Ill-Posed Hyperbolic Problem and the Uniqueness of the Solution by a Wavelet Galerkin Method

José Roberto Linhares de Mattos & Ernesto Prado Lopes

Anal. Theory Appl., 28 (2012), pp. 125-134.

Published online: 2012-06

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  • Abstract

We consider the problem $K(x)u_{xx} = u_{tt}$, $0 < x < 1$, $t \geq 0$, 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 $0 < \alpha \leq K(x) < +\infty$. 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.

  • AMS Subject Headings

65T60

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COPYRIGHT: © Global Science Press

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@Article{ATA-28-125, author = {José Roberto Linhares de Mattos , 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 = {

We consider the problem $K(x)u_{xx} = u_{tt}$, $0 < x < 1$, $t \geq 0$, 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 $0 < \alpha \leq K(x) < +\infty$. 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.3969/j.issn.1672-4070.2012.02.003}, url = {http://global-sci.org/intro/article_detail/ata/4549.html} }
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 -

We consider the problem $K(x)u_{xx} = u_{tt}$, $0 < x < 1$, $t \geq 0$, 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 $0 < \alpha \leq K(x) < +\infty$. 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.

José Roberto Linhares de Mattos , and Prado Lopes , Ernesto. (2012). Regularization of an Ill-Posed Hyperbolic Problem and the Uniqueness of the Solution by a Wavelet Galerkin Method. Analysis in Theory and Applications. 28 (2). 125-134. doi:10.3969/j.issn.1672-4070.2012.02.003
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