Methods for Solving Incorrectly Posed Problems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
1984
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f € F, (1) Au = f, where A: DA C U + F is an operator with a non-empty domain of definition D , in a metric space U, with range in a metric space F. The metrics A on U and F will be denoted by P and P ' respectively. Relative u F to the twin spaces U and F, J. Hadamard P-06] gave the following definition of correctness: the problem (1) is said to be well-posed (correct, properly posed) if the following conditions are satisfied: (1) The range of the value Q of the operator A coincides with A F ("solvability" condition); (2) The equality AU = AU for any u ,u € DA implies the I 2 l 2 equality u = u ("uniqueness" condition); l 2 (3) The inverse operator A-I is continuous on F ("stability" condition). Any reasonable mathematical formulation of a physical problem requires that conditions (1)-(3) be satisfied. That is why Hadamard postulated that any "ill-posed" (improperly posed) problem, that is to say, one which does not satisfy conditions (1)-(3), is non-physical. Hadamard also gave the now classical example of an ill-posed problem, namely, the Cauchy problem for the Laplace equation |
| Beschreibung: | 1 Online-Ressource (257p) |
| ISBN: | 9781461252801 9780387960593 |
| DOI: | 10.1007/978-1-4612-5280-1 |
Internformat
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| 500 | |a Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f € F, (1) Au = f, where A: DA C U + F is an operator with a non-empty domain of definition D , in a metric space U, with range in a metric space F. The metrics A on U and F will be denoted by P and P ' respectively. Relative u F to the twin spaces U and F, J. Hadamard P-06] gave the following definition of correctness: the problem (1) is said to be well-posed (correct, properly posed) if the following conditions are satisfied: (1) The range of the value Q of the operator A coincides with A F ("solvability" condition); (2) The equality AU = AU for any u ,u € DA implies the I 2 l 2 equality u = u ("uniqueness" condition); l 2 (3) The inverse operator A-I is continuous on F ("stability" condition). Any reasonable mathematical formulation of a physical problem requires that conditions (1)-(3) be satisfied. That is why Hadamard postulated that any "ill-posed" (improperly posed) problem, that is to say, one which does not satisfy conditions (1)-(3), is non-physical. Hadamard also gave the now classical example of an ill-posed problem, namely, the Cauchy problem for the Laplace equation | ||
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Datensatz im Suchindex
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| author | Morozov, V. A. |
| author_facet | Morozov, V. A. |
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| author_sort | Morozov, V. A. |
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| dewey-full | 518 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518 |
| dewey-search | 518 |
| dewey-sort | 3518 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4612-5280-1 |
| format | Electronic eBook |
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| spelling | Morozov, V. A. Verfasser aut Methods for Solving Incorrectly Posed Problems by V. A. Morozov New York, NY Springer New York 1984 1 Online-Ressource (257p) txt rdacontent c rdamedia cr rdacarrier Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f € F, (1) Au = f, where A: DA C U + F is an operator with a non-empty domain of definition D , in a metric space U, with range in a metric space F. The metrics A on U and F will be denoted by P and P ' respectively. Relative u F to the twin spaces U and F, J. Hadamard P-06] gave the following definition of correctness: the problem (1) is said to be well-posed (correct, properly posed) if the following conditions are satisfied: (1) The range of the value Q of the operator A coincides with A F ("solvability" condition); (2) The equality AU = AU for any u ,u € DA implies the I 2 l 2 equality u = u ("uniqueness" condition); l 2 (3) The inverse operator A-I is continuous on F ("stability" condition). Any reasonable mathematical formulation of a physical problem requires that conditions (1)-(3) be satisfied. That is why Hadamard postulated that any "ill-posed" (improperly posed) problem, that is to say, one which does not satisfy conditions (1)-(3), is non-physical. Hadamard also gave the now classical example of an ill-posed problem, namely, the Cauchy problem for the Laplace equation Mathematics Numerical analysis Numerical Analysis Mathematik Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd rswk-swf Stabilität (DE-588)4056693-6 gnd rswk-swf Inkorrekt gestelltes Problem (DE-588)4186951-5 s Partielle Differentialgleichung (DE-588)4044779-0 s 1\p DE-604 Stabilität (DE-588)4056693-6 s 2\p DE-604 https://doi.org/10.1007/978-1-4612-5280-1 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Morozov, V. A. Methods for Solving Incorrectly Posed Problems Mathematics Numerical analysis Numerical Analysis Mathematik Partielle Differentialgleichung (DE-588)4044779-0 gnd Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd Stabilität (DE-588)4056693-6 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4186951-5 (DE-588)4056693-6 |
| title | Methods for Solving Incorrectly Posed Problems |
| title_auth | Methods for Solving Incorrectly Posed Problems |
| title_exact_search | Methods for Solving Incorrectly Posed Problems |
| title_full | Methods for Solving Incorrectly Posed Problems by V. A. Morozov |
| title_fullStr | Methods for Solving Incorrectly Posed Problems by V. A. Morozov |
| title_full_unstemmed | Methods for Solving Incorrectly Posed Problems by V. A. Morozov |
| title_short | Methods for Solving Incorrectly Posed Problems |
| title_sort | methods for solving incorrectly posed problems |
| topic | Mathematics Numerical analysis Numerical Analysis Mathematik Partielle Differentialgleichung (DE-588)4044779-0 gnd Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd Stabilität (DE-588)4056693-6 gnd |
| topic_facet | Mathematics Numerical analysis Numerical Analysis Mathematik Partielle Differentialgleichung Inkorrekt gestelltes Problem Stabilität |
| url | https://doi.org/10.1007/978-1-4612-5280-1 |
| work_keys_str_mv | AT morozovva methodsforsolvingincorrectlyposedproblems |