American Mathematical Society
TRANSLATIONS Series 2 • Volume 130
One-Dimensional Inverse Problems of Mathematical Physics
aM ° American Mathematical Society „ m
One-Dimensional Inverse Problems of Mathematical Physics
American Mathematical Society
TRANSLATIONS Series 2 • Volume 130
One-Dimensional
Inverse Problems
of Mathematical Physics
By M. M. Lavrent'ev K. G. Reznitskaya V. G. Yakhno
d, o n American Mathematical Society r, --1 Providence, Rhode Island
http://dx.doi.org/10.1090/trans2/130
Translated by J. R. SCHULENBERGER
Translation edited by LEV J. LEIFMAN
1980 Mathematics Subject Classification (1985 Revision): Primary 35K05, 35L05, 35R30.
Abstract. Problems of determining a variable coefficient and right side for hyperbolic and
parabolic equations on the basis of known solutions at fixed points of space for all times
are considered in this monograph. Here the desired coefficient of the equation is a
function of only one coordinate, while the desired right side is a function only of time. On
the basis of solution of direct problems the inverse problems are reduced to nonlinear
operator equations for which uniqueness and in some cases also existence questions are
investigated. The problems studied have applied importance, since they are models for
interpreting data of geophysical prospecting by seismic and electric means.
The monograph is of interest to mathematicians concerned with mathematical physics.
Bibliography: 75 titles.
Library of Congress Cataloging-in-Publication Data
Lavrent'ev, M. M. (Mlkhail Milchanovich)
One-dimensional inverse problems of mathematical physics.
(American Mathematical Society translations; ser. 2, v. 130)
Translation of: Odnomernye obratnye zadachi matematicheskoi
Bibliography: p. 67.
1. Inverse problems (Differential equations) 2. Mathematical physics. I. Reznitskaya,
K. G. II. Yakhno, V. G. III. Title. IV. Series.
QA3.A572 vol. 130 510 s [530.1'5535] 86-7917
[QC20.7.D5]
ISBN 0-8218-3099-6
ISSN 0065-9290
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10 9 8 7 6 5 4 3 2 96 95 94 93 92
Contents
Introduction 1
Chapter I. SOLUTIONS OF DIRECT AND INVERSE PROBLEMS AND SOME OF THEIR
RELATIONS
§1. Convolution formulas 5
§2. Connections between solutions of second-order equations of various
types 6
§3. Direct and inverse problems for the heat equation and the method of
incomplete separation of variables 8
Chapter II. SOURCE PROBLEMS
§1. A linearized formulation of the problem of determining q(z) and f(t);
the case µ(z) = 1 15
§2. The linearized problem. The case of a coefficient of the leading deriva-
tive 18
§3. The problem in the exact formulation; the case p.(z) = 1 19
Chapter III. A ONE-DIMENSIONAL INVERSE PROBLEM FOR THE WAVE EQUATION
§1. Generalized solutions of boundary value problems for the wave equa-
tion 27
§2. The concept of a solution of an inverse problem in the case of
information given on a finite segment 33
§3. "Local" existence of a unique solution of the inverse problem with a
"distributed" source 34
§4. A method of constructing a "global" solution of the inverse problem
with a "distributed" source 38
§5. Uniqueness and stability of the solution of the inverse problem with a
"distributed" source 45
§6. Uniqueness of the solution of the inverse problem with a source of
perturbation concentrated at a point 52
Vi CONTENTS
Appendix
§1. The Laplace transform 59 §2. The exponential Fourier transform 60
§3. The Fourier cosine transform 61
§4. The method of spectral theory of second-order ordinary differential
operators 62
§5. Connections between solutions of linear differential equations in Banach
spaces 64
Bibliography 67
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ISBN: 0-8218-3099-6
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