TRANSLATIONS. One-Dimensional Inverse Problems of Mathematical Physics. American Mathematical Society. Series 2 Volume 130

Transcription

1 American Mathematical Society TRANSLATIONS Series 2 Volume 130 One-Dimensional Inverse Problems of Mathematical Physics am American Mathematical Society m

2 One-Dimensional Inverse Problems of Mathematical Physics

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4 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

5 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 Inverse problems (Differential equations) 2. Mathematical physics. I. Reznitskaya, K. G. II. Yakhno, V. G. III. Title. IV. Series. QA3.A572 vol s [530.1'5535] [QC20.7.D5] ISBN ISSN 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 publication (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 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 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

6 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) = The linearized problem. The case of a coefficient of the leading derivative 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 equation The concept of a solution of an inverse problem in the case of information given on a finite segment "Local" existence of a unique solution of the inverse problem with a "distributed" source A method of constructing a "global" solution of the inverse problem with a "distributed" source Uniqueness and stability of the solution of the inverse problem with a "distributed" source Uniqueness of the solution of the inverse problem with a source of perturbation concentrated at a point 52

7 Vi CONTENTS Appendix 1. The Laplace transform The exponential Fourier transform The Fourier cosine transform The method of spectral theory of second-order ordinary differential operators Connections between solutions of linear differential equations in Banach spaces 64 Bibliography 67

8 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 (Russian) R. Zh. Mat. 1968, , Some inverse problems in wave propagation theory. I, Izv. Akad. Nauk SSSR Ser. Geofiz. 1962, ; English transl. in Bull. (Izv.) Acad. Sci. USSR Geophys. Ser 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), (Russian) R. Zh. Mat. 19'73, Yu. E. Anikonov, Some methods for the study of multidimensional inverse problems for differential equations, "Nauka", Novosibirsk, (Russian) 5. A. Erdelyi et al., Tables of integral transforms, Vol. 1, McGraw-Hill, B. van der Pol and H. Bremmer, Operational calculus, based on the two-sided Laplace integral, Cambridge Univ. Press, S. P. Belinskil, On an inverse problem for linear t-hyperbolic systems of first-order equations, Mat. Problemy Geofiz. Vyp. 6 (1975), part 2, (Russian) Zbl. 357 # Yu. M. Berezanskii, Expansion in eigenfunctions of selfadjoint operators, "Naukova Dumka", Kiev, 1965; English transl., Amer. Math. Soc., Providence, R. I., A. S. Blagoveshchenskii, The inverse problem in the theory of seismic wave propagation, Problemy Mat. Fiz., vyp. 1, Izdat. Leningrad. Univ., Leningrad, 1966, pp ; English transl. in Topics in Math. Phys., no. 1, Plenum Press, New York, 1967; errata, ibid, vyp. 3, 1968, p. 103 = no. 3, , 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 ; English transl. in Topics in Math. Phys., no. 4, Plenum Press, New York, , 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, Herbert Gajewski, Konrad GrOger and Klaus Zacharias, Nichtlineare Operatorgleichungen and Operatordifferentialgleichungen, Akademie-Verlag, Berlin, M. G. Gasymov, Analytic properties of the spectral functions of a selfadjoint Sturm-Liouville operator, Dokl. Akad. Nauk SSSR 150 (1963), ; 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), ; 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, (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.

9 68 BIBLIOGRAPHY 18. V. A. Ditkin and P. I. Kuznetsov, Handbook of operational calculus, GITTL, Moscow, (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, 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 (Russian) Zbl. 518 # , A finite-difference inverse problem for the wave equation, Questions of the Well-posedness of Problems of Mathematical Physics (M. M. Lavrent'ev, editor), Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1977, pp (Russian) R. Zh. Mat # , 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), (Russian) 24. R. Courant, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience M. M. Lavrent'ev and K. G. Reznitskaya, Uniqueness theorems for certain nonlinear inverse problems for parabolic equations, Dokl. Akad. Nauk SSSR 208 (1973), ; English transl. in Soviet Math. Dokl. 14 (1973). 26, On the uniqueness of a solution of an inverse problem in the theory of heat conduction, Mat. Problemy Geofiz. Vyp. 6 (1975), part 1, (Russian) Zbl. 323 # , An inverse problem with unknown source, Dokl. Alad. Nauk SSSR 246 (1979), ; English transl. in Soviet Math. Dokl. 20 (1979). 28., An inverse problem with an unknown source, Uniqueness, Stability and Methods of Solving Inverse and Ill-posed Problems (M. M. Lavrent'ev, editor), Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1980, pp (Russian) Zbl. 464 # M. M. Lavrent'ev, V. G. Romanov and G. P. Shishat sldi, Ill-posed problems of mathematical physics and analysis, "Nauka", Moscow, 1980; English transl., Amer. Math. Soc., Providence, R. I., A. Ladyzhenskaya, Boundary value problems of mathematical physics, "Nauka", Moscow, (Russian) 31. B. M. Levitan, On a theorem of expansion in eigenfunctions of second-order differential equations, Dokl. Akad. Nauk SSSR 71 (1950), (Russian) 32., Expansion in eigenfunctions of second-order differential equations, GITTL, Moscow, (Russian) 33. V. A. Marchenko, Some questions on the theory of a second order differential operator, Dokl. Akad. Nauk SSSR 72 (1950), (Russian) 34., Spectral theory of Sturm-Liouville operators, "Naukova Dumka", Kiev, (Russian) 35., Sturm-Liouville operators and their applications, "Naukova Dumka", Kiev, (Russian) 36. G. I. Marchuk, Methods of numerical mathematics, 2nd ed., " Nauka", Moscow, 1980; English transl., Springer-Verlag, G. Myuntts [Ch. H. Mintz], Integral equations. Vol. I, ONTI, Moscow, (Russian) 38. M. A Naimark, Linear differential operators, 2nd ed., "Nauka", Moscow, 1969; English transl. of 1st ed., Parts I, II, Ungar, New York, 1967, L. P. Nizhnik, The inverse time-dependent scattering problem, "Naukova Dumka", Kiev, (Russian) 40. A. F. Nikiforov and V. B. Uvarov, Foundations of the theory of special functions, "Nauka", Moscow, 1974; French transl., " Mir", Moscow, K. G. Reznitskaya, The connection between solutions of the Cauchy problem for equations of different types and inverse problems, Mat. Problemy Geofiz. Vyp. 5 (1974), part 1, (Russian) Zbl. 328 # , The source problem. The linearized case, Questions of the Well-posedness of Problems of Mathematical Physics (M. M. Lavrent'ev, editor), Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1977, pp (Russian) R. Zh. Mat # 12E706.

10 BIBLIOGRAPHY V. G. Romanov, A uniqueness theorem for a one-dimensional inverse problem for the wave equation, Mat. Problemy Geofiz. Vyp. 2 (1971), (Russian) R. Zh. Mat. 1971, , The problem of determining the one-dimensional propagation speed of signals in a half-space on the basis of the regime of oscillations of a point of this space, Mat. Problemy Geofiz. Vyp. 3 (1972), (Russian) R. Zh. Mat. 1973, , Inverse problems for differential equations, Novosibirsk. Gos. Univ., Novosibirsk, (Russian) 46., A formulation of an inverse problem for first order symmetric hyperbolic systems, Mat. Zametki, 24 (1978), ; English transl. in Math. Notes 24 (1978). 47., An inverse problem on the propagation of electric oscillations in wires, Mat. Problemy Geofiz. Vyp. 1 (1969), (Russian) R. Zh. Mat. 1970, , A one-dimensional inverse problem for the wave equation, Dokl. Akad. Nauk SSSR 211 (1973), ; English transl. in Soviet Phys. Dokl. 18 (1973/74). 49., A formulation of the inverse problem for the generalized wave equation, Dokl. Akad. Nauk SSSR 181 (1968), ; English transl. in Soviet Math. Dokl. 9 (1968). 50., The one-dimensional inverse problem for the telegraph equation, Differentsial'nye Uravneniya 4 (1968), ; English transl. in Differential Equations 4 (1968). 51., Some inverse problems for equations of hyperbolic type, "Nauka", Novosibirsk, 1969; English transl., Integral geometry and inverse problems for hyperbolic equations, Springer-Verlag, , Inverse problems for differential equations. Inverse kinematic problems of seismology, Novosibirsk. Gos. Univ., Novosibirsk, (Russian) 53., Inverse problems for hyperbolic equations and energy inequalities, Dokl. Akad. Nauk SSSR 242 (1978), ; English transl. in Soviet Math. Dokl. 19 (1978). 54. E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, A. N. Tikhonov, On the uniqueness of the solution of the problem of electric prospecting, Dokl. Akad. Nauk SSSR 69 (1949), (Russian) 56. L. D. Faddeev, The inverse problem in quantum scattering theory. I, Uspekhi Mat. Nauk 14 (1959), no. 4 (88), ; English transl., J. Mathematical Phys. 4 (1963), , The inverse problem in quantum scattering theory. II, Itogi Nauki: Sovremennye Problemy Mat., vol. 3, VINITI, Moscow, 1974, pp ; English transl. in J. Soviet Math. 5 (1976), no William Feller, An introduction to probability theory and its applications. Vol. II, Wiley, K. Chadan and P. C. Sabatier, Inverse problems in quantum scattering theory, Springer-Verlag, V. G. Yakhno, The inverse problem for the wave equation. Uniqueness and stability theorems, Preprint No. 206, Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, (Russian) R. Zh. Mat. 1980, , The inverse problem for the wave equation (necessary and sufficient conditions for the existence of a solution), Uniqueness, Stability and Methods of Solving Inverse and Ill-posed Problems (M. M. Lavrent'ev, editor), Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1980, pp (Russian) Zbl. 464 # , The inverse problem for the wave equation (construction of a solution of the inverse problem), Preprint No. 235, Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, (Russian) R. Zh. Mat. 1981, , On the problem of determining the coefficient of a hyperbolic equation, All-Union Conf. Ill-posed Problems, Abstracts of Reports, "Ilim", Frunze, 1979, pp (Russian) 64., One-dimensional inverse problem for the wave equation, Dokl. Akad. Nauk SSSR 255 (1980), ; English transl. in Soviet Math. Dokl. 22 (1980). 65., An iterative method for constructing the solution of an inverse problem, Conditionally Well-posed Mathematical Problems and Problems of Geophysics (M. M. Lavrent'ev, editor), Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1979, pp (Russian) Zbl. 501 # G. N. Watson, A treatise on the theory of Besse! functions, 2nd ed., Cambridge Univ. Press, Cambridge, and Macmillan, New York, V. Ambarzumian, Uber eine Frage der Eigenwerttheorie, Z. Phys. 53 (1929), E. Wiechert and K. Zoeppritz, Uber Erdbebenwellen, Nachr. KOnigl. Ges. Wiss. Gottingen Math.-Phys. Kl. 1907,

11 70 BIBLIOGRAPHY 69. L. R. Bragg and J. W. Dettman, An operator calculus for related partial differential equations, J. Math. Anal. Appl. 22 (1968), , A class of related Dirichlet and initial value problems, Proc. Amer. Math. Soc. 21 (1969), , Related problems in partial differential equations, Bull. Amer. Math. Soc. 74 (1968), G. Herglotz, &her die Elastizitat der Erde hei Berlicksichtigung ihrer variablen Dichte, Z. Math. and Phys. 52 (1905), Emile Picard, Lecons sur quelques equations fonctionnelles avec applications a divers problemes d 'analyse et de physique mathematique, Gauthier-Villars, Paris, L. Schwartz, Theorie des distributions, Vols. I, II, Actualites Sci. Indust., nos. 1091, 1122, Hermann, Paris, 1950, S. G. Krein, Linear differential equations in Banach space, "Nauka", Moscow, 1967; English transl., Amer. Math. Soc., Providence, R. I., 1971.

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