| Methods for Solving Incorrectly Posed Problems Softcover Repri Edition Contributor(s): Morozov, V. a. (Author), Nashed, Z. (Editor), Aries, A. B. (Translator) |
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ISBN: 0387960597 ISBN-13: 9780387960593 Publisher: Springer OUR PRICE: $52.24 Product Type: Paperback Published: November 1984 |
| Additional Information |
| BISAC Categories: - Mathematics | Number Systems - Mathematics | Numerical Analysis |
| Dewey: 515.353 |
| LCCN: 84013961 |
| Physical Information: 0.59" H x 6.14" W x 9.21" (0.87 lbs) 257 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |