Fifth Mississippi State Conference on Differential Equations and Computational Simulations,
Electron. J. Diff. Eqns., Conf. 10, 2002, pp. 211-225.

A wavelet Galerkin method applied to partial differential equations with variable coefficients

Jose Roberto Linhares de Mattos & Ernesto Prado Lopes

We consider the problem $K(x)u_{xx}=u_{t}$ , $0 less than x less than 1$, $t\geq 0$, where $K(x)$ is bounded below by a positive constant. The solution on the boundary $x=0$ is a known function $g$ and $u_{x}(0,t)=0$. This is an ill-posed problem in the sense that a small disturbance on the boundary specification $g$, can produce a big alteration on its solution, if it exists. We consider the existence of a solution $u(x,\cdot)\in L^{2}(R)$ and we use a wavelet Galerkin method with the Meyer multi-resolution analysis, to filter away the high-frequencies and to obtain well-posed approximating problems in the scaling spaces $V_{j}$. We also derive an estimate for the difference between the exact solution of the problem and the orthogonal projection, onto $V_{j}$, of the solution of the approximating problem defined in $V_{j-1}$.

Published February 28, 2003.
Subject classifications: 65T60.
Key words: Wavelet, multi-resolution analysis.

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Note: The authors were nitially posted in reversed.

Jose Roberto Linhares de Mattos
Federal University Fluminense
Institute of Mathematics, Department of Geometry
Rua Mario Santos Braga, s/n, Campus do Valonguinho
Niteroi, RJ, CEP 24020-140, Brazil
Ernesto Prado Lopes
Federal University of Rio de Janeiro
COPPE, Systems and Computing Engineering Program
Tecnology Center, Bloco H
Institute of Mathematics, Tecnology Center, Bloco C
Ilha do Fundao, Rio de Janeiro RJ, CEP 21945-970, Brazil

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