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American Mathematical Society TRANSLATIONS Series 2 • Volume 130 American Mathematical Society TRANSLATIONS Series 2 • Volume 130 One-Dimensional Inverse Problems of Mathematical Physics aM „ °m American Mathematical Society One-Dimensional Inverse Problems of Mathematical Physics http://dx.doi.org/10.1090/trans2/130 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 do, n American Mathematical Society r, --1 Providence, Rhode Island 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 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this pub- lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright Q1986 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 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 Bibliography* 1. A. S. Alekseev, Inverse dynamical problems of seismology, Some Methods and Algorithms for the Interpretation of Geophysical Data (M. M. Lavrent'ev, editor), "Nauka," Moscow, 1967, pp. 9-84. (Russian) R. Zh. Mat. 1968, 65562. 2. , Some inverse problems in wave propagation theory. I, Izv. Akad. Nauk SSSR Ser. Geofiz. 1962, 1514-1522; English transl. in Bull. (Izv.) Acad. Sci. USSR Geophys. Ser. 1962. 3. A. S. Alekseev and A. G. Megrabov, The direct and inverse problems of the scattering of plane waves by inhomogeneous transition layers, Mat. Problemy Geofiz. Vyp. 3 (1972), 8-36. (Russian) R. Zh. Mat. 19'73, 55442. 4. Yu. E. Anikonov, Some methods for the study of multidimensional inverse problems for differential equations, "Nauka", Novosibirsk, 1978. (Russian) 5. A. Erdelyi et al., Tables of integral transforms, Vol. 1, McGraw-Hill, 1954. 6. B. van der Pol and H. Bremmer, Operational calculus, based on the two-sided Laplace integral, Cambridge Univ. Press, 1950. 7. S. P. Belinskil, On an inverse problem for linear t-hyperbolic systems of first-order equations, Mat. Problemy Geofiz. Vyp. 6 (1975), part 2, 100-109. (Russian) Zbl. 357 #35077. 8. Yu. M. Berezanskii, Expansion in eigenfunctions of selfadjoint operators, "Naukova Dumka", Kiev, 1965; English transl., Amer. Math. Soc., Providence, R. I., 1968. 9. A. S. Blagoveshchenskii, The inverse problem in the theory of seismic wave propagation, Problemy Mat. Fiz., vyp. 1, Izdat. Leningrad. Univ., Leningrad, 1966, pp. 68-81; English transl. in Topics in Math. Phys., no. 1, Plenum Press, New York, 1967; errata, ibid, vyp. 3, 1968, p. 103 = no. 3, 1969. 10. , A one-dimensional inverse boundary value problem for a second order hyperbolic equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 15 (1969), 85-90; English transl. in Sem. Math. V. A. Steklov Math. Inst. Leningrad 15 (1969). 11. , The various formulations of the one-dimensional inverse problem for the telegraph equation, Problemy Mat. Fiz., vyp. 4, Izdat. Leningrad. Univ., Leningrad, 1970, pp. 40-41; English transl. in Topics in Math. Phys., no. 4, Plenum Press, New York, 1971. 12. , On a local method of solving a nonstationary inverse problem for a nonhomogeneous string, Trudy Mat. Inst. Steklov. 115 (1971), 28-38; English transl. in Proc. Steklov Inst. Math. 115 (1971). 13. V. S. Vladimirov, Generalized functions in mathematical physics, "Nauka", Moscow, 1976; English transl. of 1979 ed., "Mir", Moscow, 1979. 14. Herbert Gajewski, Konrad GrOger and Klaus Zacharias, Nichtlineare Operatorgleichungen and Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974. 15. M. G. Gasymov, Analytic properties of the spectral functions of a selfadjoint Sturm-Liouville operator, Dokl. Akad. Nauk SSSR 150 (1963), 971-974; English transl. in Soviet Math. Dokl. 4 (1963). 16. I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 309-360; English transl. in Amer. Math. Soc. Transl. (2) 1 (1955). 17. S. V. Gol'din, Interpretation of data in the seismic method of reflected waves, "Nedra", Moscow, 1979. (Russian) * Editor's note. The periodical Mat. Problemy Geofiz., frequently cited herein, is edited by M. M. Lavrent'ev and A. S. Alekseev, and published by the Computing Center, Siberian Academy of Sciences of the USSR, Novosibirsk. 68 BIBLIOGRAPHY 18. V. A. Ditkin and P. I. Kuznetsov, Handbook of operational calculus, GITTL, Moscow, 1951. (Russian) 19. V. A. Ditkin and A. P. Prudnikov, Integral transforms and operational calculus, 2nd aug. ed., "Nauka", Moscow, 1974; French transl., "Mir", Moscow, 1978; English transl. of 1st ed., Pergamon Press, 1965. 20. S. I. Kabanikhin, On the solvability of the inverse dynamical problem of seismology, Conditionally Well-posed Mathematical Problems and Problems of Geophysics (M. M. Lavrent'ev, editor), Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1979, pp. 43-51. (Russian) Zbl. 518 # 35078. 21. , A finite-difference inverse problem for the wave equation, Questions of the Well-posed- ness of Problems of Mathematical Physics (M. M. Lavrent'ev, editor), Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1977, pp. 57-69. (Russian) R. Zh. Mat. 1978 # 1051198. 22. , Application of energy inequalities of an inverse problem for a hyperbolic equation, Differentsial'nye Uravneniya 15 (1979), 61-67; English transl. in Differential Equations 15 (1979). 23. M. G. Krein, On a method of effective solution of an inverse boundary value problem, Dokl. Akad. Nauk SSSR 94 (1954), 987-990. (Russian) 24. R. Courant, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience 1962. 25. M. M. Lavrent'ev and K. G. Reznitskaya, Uniqueness theorems for certain nonlinear inverse problems for parabolic equations, Dokl. Akad. Nauk SSSR 208 (1973), 531-532; English transl. in Soviet Math. Dokl.
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