Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis: Proceedings of the International Conference, Samarkand, Uzbekistan [Reprint 2014 ed.] 9783110936520, 9783110364170

Table of contents :
Problems of integral geometry on curves and surfaces in Euclidean space
Existence of solutions of the first boundary-value problem for the third order equations of mixed type in unbounded domain
On the monotone error rule for choosing the regularization parameter in ill-posed problems
Well-posedness of one-dimensional inverse acoustic problem in L2 for small depth or small data
Fourier series in Banach spaces
Ill-posed and inverse problems for hyperbolic equations
Systems of linear integral equations of Volterra type with singular and super-singular kernels
Inverses of a family of bounded linear operators, generalized pythagorean theorems and reproducing kernels
Cauchy problem for the Helmholtz equation
CONTRIBUTED PAPERS
On problem for a third order equation with multiple characteristics
On uniqueness and stability of solution of the Cauchy problem for pseudoparabolic equation
Uniqueness theorem for an unbounded domain
Recovery of a function set by integrals along a curve family in the plane
Uniqueness of extension of solutions of differential equations of the second order

Citation preview

INVERSE A N D ILL-POSED PROBLEMS SERIES

Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis

Also available in the Inverse and Ill-Posed Problems Series: Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations AG. Megrabov Nonclassical LinearVolterra Equations of the First Kind AS. Apartsyn Poorly Visible Media in X-ray Tomography D.S. Anikonov, V.G. Nazarov, and I.V. Prokhorov Dynamical Inverse Problems of Distributed Systems V.l. Maksimov Theory of Linear Ill-Posed Problems and its Applications V.K. Ivanov.V.V.Vasin andV.P.Tanana Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation MAAtakhodzhaev Investigation Methods for Inverse Problems V.G. Romanov Operator Theory. Nonclassical Problems S.G. Pyatkov Inverse Problems for Partial Differential Equations Yu.Ya. Belov Method of Spectral Mappings in the Inverse Problem Theory V.Yurko Theory of Linear Optimization I.I. Eremin Integral Geometry and Inverse Problems for Kinetic Equations AKh.Amirov Computer Modelling in Tomography and Ill-Posed Problems M.M. Lavrent'ev, S.M.Zerkal and O.ETrofimov An Introduction to Identification Problems via Functional Analysis A. Lorenzi Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev Inverse Problems for Kinetic and Other Evolution Equations Yu.E Anikonov Inverse Problems ofWave Processes AS. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S.P. Shishatskii, A. Asanov and ER. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov

Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev Introduction to theTheory of Inverse Problems A L Bukhgeim Identification Problems ofWave Phenomena Theory and Numerics S.I. Kabanikhin and A. Lorenzi Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A.I. Kozhanov Inverse Problems ofVibrational Spectroscopy A.G.Yagola, I.V. Kochikov, G.M. Kuramshina andYuA Pentin Elements of theTheory of Inverse Problems A.M. Denisov Volterra Equations and Inverse Problems A.L Bughgeim Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P.Tanana Inverse and Ill-Posed Sources Problems Yu.E. Anikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P.Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A.Asanov and EUAtamanov Formulas in Inverse and Ill-Posed Problems Yu.E.Anikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.E Anikonov Ill-Posed Problems with A Priori Information V.V.Vasin andA.LAgeev Integral Geometry ofTensor Fields VA. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis PROCEEDINGS OF THE INTERNATIONAL CONFERENCE SAMARKAND, UZBEKISTAN

Editor-in-Chief: M.M. Lavrent'ev Managing Editor: S.Í. Kabanikhin Editorial Board: Akb.H. Begmatov, T.D. Djuraev, S. Saitoh and N[. Yamamoto

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2003

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INVITED PAPERS

Problems of integral geometry on curves and surfaces in Euclidean space A. H. Begmatov and A. H. Begmatov

1

Existence of solutions of the first boundary-value problem for the third order equations of mixed type in unbounded domain T. D. Djuraev and A. R. Hashimov

19

On the monotone error rule for choosing the regularization parameter in ill-posed problems U. Hämarik and U. Tautenhahn

27

Well-posedness of one-dimensional inverse acoustic problem in L2 for small depth or small data S. I. Kabanikhin, Κ. T. Iskakov, and M. Yamamoto Fourier series in Banach spaces Dj. Khadjiev and A. Çavu§ Ill-posed and inverse problems for hyperbolic equations M. M. Lavrent'ev Systems of linear integral equations of Volterra type with singular and super-singular kernels N. Rajabov Inverses of a family of bounded linear operators, generalized pythagorean theorems and reproducing kernels S. Saitoh Cauchy problem for the Helmholtz equation Sh. Yarmukhamedov and I. Yarmukhamedov

57 71 81

103

125 143

CONTRIBUTED PAPERS

On problem for a third order equation with multiple characteristics S. Abdinazarov and B. M. Kholboev On uniqueness and stability of solution of the Cauchy problem for pseudoparabolic equation B. K. Amonov and S. S. Kobilov Uniqueness theorem for an unbounded domain Z. R. Ashurova and Yu. J. Zhuraev Recovery of a function set by integrals along a curve family in the plane A. H. Begmatov and Z. H. Ochilov Uniqueness of extension of solutions of differential equations of the second order A. Haidarov and D. Shodiev

173

179 185

191

199

This book is the Proceedings of the International Conference "Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis" which was held at the Samarkand State University, Samarkand, Uzbekistan from 11 to 15 September 2000. The Conference was organized jointly by the Samarkand State University and Sobolev Institute of Mathematics, Novosibirsk, Russia. Advice and general guidance were provided by the International Programme Comittee. More than 90 participants from Germany, Japan, Kazakhstan, Russia, Tajikistan, Turkey, and Uzbekistan presented their lectures at the Conference. The scientific program of the Conference covered the following topics: • Theory of 111-Posed Problems • Inverse Problems for Differential Equations • Boundary Value Problems for Equations of Mixed Type • Integral Geometry • Mathematical Modelling and Numerical Methods in Natural Sciences The Proceedings bring together fundamental research articles in the major areas of the numerated fields of analysis and mathematical physics. The papers in this volume represent all plenary and some contributed lectures presented at the conference. All the papers have undergone peer review. We would like to thank the Conference paricipants for their interesting reports and stimulating discussions. We express our sincere thanks to the authors for submitting articles of such high quality and to the referees for their thoughtful reviews. Finally, we would like to thank the staff at VSP for their help in publishing the Proceedings.

M. M. Lavrent 'ev S. I.

Kabanikhin

Akb. H. T. D. S. M.

Begmatov Djuraev

Saitoh Yamamoto

El-Posed and Non-Classical Problems of Mathematical Physics and Analysis, Samarkand, 2000, pp. 1-18 M.M. Lavrent'ev and S.I. Kabanikhin (Eds) © VSP 2002

Problems of integral geometry on curves and surfaces in Euclidean space Akbar H. BEGMATOV* and Akram H. BEGMATOV*

1.

INTRODUCTION

Integral geometry studies the transformations assigning to functions on a manifold X their integrals over submanifolds from certain set M [23]. This important and intensively developing domain of modern mathematics is closely connected with the theory of partial differential equations, mathematical physics, geometric analysis. This direction has many applications in mathematical study of problems of seismic exploration, interpretation of the data of geophysical and aerospace observations, in solving inverse problems of astrophysics and hydroacoustics (see [33] and the references given there). Methods developed here are basic for solving the problems of medical and industrial tomography [17, 18, 34]. The problem of recovery of a function from its integrals over all possible hyperplanes in Euclidean space was considered by J. Radon [37]. In this classical work the explicit inversion formulas for even and odd-dimensional spaces were obtained. Also, the methods of solution of such problems were developed and the various analogs of this integral transformation were considered [37]. Note that P. Funk [21], the year before, had investigated the problem of recovery of even function in the sphere from its the integralsover big circles. 'Novosibirsk, Sobolev Institute of Mathematics. E-mail: [email protected] * Samarkand State University, Uzbekistan. E-mail: [email protected] The work was partially supported by State Commitee of Science and Technology of Uzbek Republic (grants Nos. 15/99 and 2/01).

2

Α. Η. Begmatov

and Α. Η.

Begmatov

The connection of problems of integral geometry with differential equations was investigated by J. John [26, 27] (see, also, [28]). The transformation (called after by the ray transformation) which assigns to a function in three three-dimensional space its integrals over all possible straight lines was considered in [27]. Basing on the theory of Lie groups, I. M. Gel'fand with co-authors had considered the problems of integral geometry on linear manifolds. This had become the basic for investigation of integral geometry for wide classes of homogeneous spaces. Interesting inversion formulas were obtained here. The problems of reconstructing a function from its integrals over surfaces of second order were also widely considered (see [33] and the bibliography cited here). The first important results for a multidimensional inverse kinematic seismic problem were obtained by M. M. Lavrent'ev and V. G. Romanov [32] by the method of its reducing to the problem of determining the function from its mean values along all possible circles with centers in a fixed line. This work brought the attention to the cases when integration goes along the manifolds of more complicated geometric structure. Namely, in [32] the connection between multidimensional inverse problems for partial differential equations and the problems of integral geometry was revealed. The manifolds which arise after reducing the inverse problems to the problems of integral geometry are naturally connected with the initial differential equation. For the equation with variable coefficients these geometric objects may be sufficiently complicated. V. G. Romanov had investigated the problems of uniqueness and stability in the case when the manifolds are invariant under the group of all motions parallel to a (n — l)-dimensional hyperplane [38, 39]. The weight functions are also assumed to be invariant under this group. The important results on uniqueness and stability of solution were obtained by Yu. E. Anikonov and A. L. Bukhgeim for the following classes of integral geometry problems: the solution is sought in the classes of functions analytic with respect to some variables; the manifolds along which the integrations goes depend analytically on some variables (parameters) (see [1, 16]). Questions of uniqueness, obtaining stability estimates and inversion formulas for various classes of problems of integral geometry in Euclidean space were investigated in [2-14], In the second section of our paper uniqueness questions for a wide class of problems of integral geometry in the plane are considered. Unlike the works mentioned above we do not impose the conditions of invariance or analyticity. The uniqueness theorem is established for smooth convex curves.

Problems of integral

geometry

3

The third section is devoted to weakly ill-posed problems of integral geometry on curves and surfaces with singularities in the vertices. The existence and uniqueness theorems and the explicit inversion formulas are obtained as well as the stability estimates in Sobolev spaces. For the problem of integral geometry on the family of cone surfaces in the even-dimensional space very simple representation of solution was obtained. On the basis of these results the stability estimates in Sobolev spaces were obtained; therefore, weak ill-posedness of the problem was established. The existence theorem was obtained also. Note that in our case (the cone surfaces) the odd case and the even case differ essentially. Such situation occurs frequently in the problems of integral geometry. Finally, in the fourth section we consider the problem of recovery of the function given the integrals of this function on a family of lines in threedimensional space which are generatrices of cones. Such integral transformations are called ray transformations [25] and they have wide applications in computer tomography [34]. The inversion formulas of the ray transformation connected with the cone scheme of scanning the computer tomography may be found in the works of H.K. Tuy [40], D.V. Finch [20], P. Grangeat [24] (see, also, the surveys of F. Natterer [35], V. P. Palamodov [36] and the references cited there). A.A. Kirillov [29], A.S. Blagoveshchenskii [15], I.M. Gel'fand and A.B. Goncharov [22] axe investigated various settings of the problems of recovery of a function from its integrals taken along the lines intersecting a certain set in the space. The explicit formulas of effective determining the desired function were obtained. The problem that we consider is connected with auxiliary problems of analytic extension. Unlike the problems considered in [11-13] this problem is strongly ill-posed. The uniqueness theorem for solution of the problem in the class of continuous finite functions is established. The estimate of conditional stability of solution of the logarithmic type problem was obtained. 2.

THE PROBLEM OF INTEGRAL GEOMETRY OF THE VOLTERRA T Y P E OF GENERAL FORM IN THE P L A N E

Let £ = (&>&),!/ = (2/1,2/2), χ = (χι,χί), R+ = {x = (xi,x2) • X2 > 0}. We consider the family of the curves {Γ(ζ)} in the strip L = {x e R+ : 0 < X2 < h, h < 00}. The curve from this set is defined by the equation £2 = Ψ(Χ, £i)· We assume also that for any χ € L and any angle a G [0, αχ] U [02, π], where

A. H. Begmatov and A. H. Begmatov

4

O < αχ < π/2 < c*2 < π there exists a unique curve from {Γ(χ)} passing through the point χ at the angle a with positive direction of the axis OX\. We consider the problem of recovery of a function of two variables u(x) if we know the integrals of this function with a given weight function p(:r, ξι) on the family {Γ(χ)}. This is the problem of solution of the operator equation in the function u(x): ί ρ(χ,ξΜΟάξι J Γ(χ)

= ί(χ).

(1)

We can represent equation (1) as follows: ¡•xi+hi / ρ(χ,ξιΗξ)άξ1=!(χ), where h\ — h\(x), h2 =

(2)

h2(x).

T h e o r e m 1. Let the right-hand side f(x) of the equation (2) be known for all χ from the strip L; hk G C 2 , k — 1,2, ρ e C3, φ G C 7 and the following conditions hold ρ(χ,ξι) φ(χ, Çi-hi)

> ¿i > 0;

= φ(χ, ξι + h2),

φ(χ,χι)

φ{χ, xi-hi)

-οο < 63 < S dtì

< ¿2
ε for all r 6 Π

T. D. Djuraev and A. R. Hashimov

22

and satisfying the condition r(0) = 0. Here the function Φ is such that the right-hand sides of the below formulas (9), (10) will be absolutely continuous. For problem (1), (2) the following theorems hold [4]. Theorem 1 [the analog of the Sen-Venan principle]. Suppose that u(x) is the generalized solution of problem (1), (2) in the domain Ω of the class ), where ( a V ) X i - (aW)XiSj

+ 3c[ j x . x . - 2aia -2c\>0

and f(x) = 0 in Ωτο. (9)

Then for each RQ and R such that 0 < Ro < R the following holds [

exp {-(R

Q{u) dx
0 sup |