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

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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