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Table of contents (5 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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 defini tion 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 ("sol vabi li ty" 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.
Authors and Affiliations
Bibliographic Information
Book Title: Methods for Solving Incorrectly Posed Problems
Authors: V. A. Morozov
DOI: https://doi.org/10.1007/978-1-4612-5280-1
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer-Verlag New York Inc. 1984
Softcover ISBN: 978-0-387-96059-3Published: 20 November 1984
eBook ISBN: 978-1-4612-5280-1Published: 06 December 2012
Edition Number: 1
Number of Pages: 257
Topics: Numerical Analysis