A wavelet Galerkin method applied to partial differential equations with variable coefficients
We consider the problem $K(x)u_{xx}=u_{t}$ , $0<x<1$, $tgeq 0$, 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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2003-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/conf-proc/10/l2/abstr.html |
| _version_ | 1818759527201243136 |
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| author | Jose Roberto Linhares De Mattos Ernesto Prado Lopes |
| author_facet | Jose Roberto Linhares De Mattos Ernesto Prado Lopes |
| author_sort | Jose Roberto Linhares De Mattos |
| collection | DOAJ |
| description | We consider the problem $K(x)u_{xx}=u_{t}$ , $0<x<1$, $tgeq 0$, 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 $V_{j-1}$. |
| first_indexed | 2024-12-18T06:44:08Z |
| format | Article |
| id | doaj.art-d68d261706054a7195e80424aaa7a0ff |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-18T06:44:08Z |
| publishDate | 2003-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-d68d261706054a7195e80424aaa7a0ff2022-12-21T21:17:32ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912003-02-01Conference10211225A wavelet Galerkin method applied to partial differential equations with variable coefficientsJose Roberto Linhares De MattosErnesto Prado LopesWe consider the problem $K(x)u_{xx}=u_{t}$ , $0<x<1$, $tgeq 0$, 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 $V_{j-1}$.http://ejde.math.txstate.edu/conf-proc/10/l2/abstr.htmlWaveletmulti-resolution analysis. |
| spellingShingle | Jose Roberto Linhares De Mattos Ernesto Prado Lopes A wavelet Galerkin method applied to partial differential equations with variable coefficients Electronic Journal of Differential Equations Wavelet multi-resolution analysis. |
| title | A wavelet Galerkin method applied to partial differential equations with variable coefficients |
| title_full | A wavelet Galerkin method applied to partial differential equations with variable coefficients |
| title_fullStr | A wavelet Galerkin method applied to partial differential equations with variable coefficients |
| title_full_unstemmed | A wavelet Galerkin method applied to partial differential equations with variable coefficients |
| title_short | A wavelet Galerkin method applied to partial differential equations with variable coefficients |
| title_sort | wavelet galerkin method applied to partial differential equations with variable coefficients |
| topic | Wavelet multi-resolution analysis. |
| url | http://ejde.math.txstate.edu/conf-proc/10/l2/abstr.html |
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