On an inverse problem for the transport equation

Аниконов, Д. С.

doi:10.1007/bf00967523

Cited by 7 publications

(11 citation statements)

References 4 publications

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“…For the first time, similar ideas were used by the author in [3,4]. Then this method was developed in the study of problems of X-ray tomography and integral geometry [5][6][7][8][9][10]. The statement of the problems with unknown boundaries is not rare these days.…”

Section: Remark 32mentioning

confidence: 97%

A Method for Studying Singular Integral Equations

Аниконов

2010

Sib Math J

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We examine a singular integral equation of the first kind on a bounded open set of an n-dimensional space. Open subsets with a common (contact) (n − 1)-dimensional piecewise smooth part of boundaries are selected. The underdetermined case is treated in which the unknown part of the integrand depends on 2n independent variables whereas a given integral depends only on n variables. In this situation we pose the problem of finding the contact part of the boundaries and prove unique solvability of the problem. We use the following notation: ρ(A, B) is the distance between sets A and B in R n ; ∂A is the boundary ofConsider a function g(x, y, ω) measurable on T ×T ×Ω and bounded almost everywhere in the following sense: There exist T and Ω ( T ∈ T and Ω ∈ Ω) coinciding in measure with T and Ω and such that |g(x, y, ω)| ≤ const for (x, y, ω) ∈ T × T × Ω. Here and in what follows, the notation const is used to denote a positive constant. In order to reduce the number of denotations, the bounded functions are often written below as O(1). We put T i = T i ∩ T and T 0 = T 1 ∪ T 2 , i = 1, 2. A point of the unit sphere Ω is denoted by ω if it designates an independent variable. If this point is expressed through other variables then we employ the letter s; for example, s = (y − x)/|y − x|. Consider the integral(1.1)The functions V i (x) for x ∈ T i , i = 1, 2, are understood to be singular integrals. We assume that T i , i = 1, 2, and g(x, y, ω) satisfy the following condition: for every u ∈ T i ,Of course, this restriction means that the integrand is a summable function. For convenience, g(x, y, ω) is assumed to be extended by zero in y beyond T . Note that if we use the spherical coordinates y − u = tν, ν ∈ Ω, dy = t n−1 dtdν then (1.2) is a consequence of the condition Ω dν ∞ 0 |g(x, u + tν, ω) − g(x, u, ω)| t dt ≤ const, ω ∈ Ω, x ∈ T .

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Section: Remark 32mentioning

confidence: 97%

A Method for Studying Singular Integral Equations

Аниконов

2010

Sib Math J

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“…In studying inverse problems for the transport equation, we considered similar questions [7,8] and came to the new statement of an integral geometry problem which is characterized by the incompleteness of data. Namely, the integrand depends not only on the integration points but also on additional parameters describing the sets over which the integration is carried out.…”

Section: Main Notations Statement Of the Problem And Auxiliariesmentioning

confidence: 99%

An integral geometry underdetermined problem for a family of curves

Аниконов

Konovalova

2015

Sib Math J

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In a general integral geometry problem, there are given the integrals of an unknown function over certain manifolds. The traditional statement of the problem consists in determining the integrand. We consider the case of an underdetermined problem when the unknown functions depend on a greater number of variables than the given integrals. These situations appear in a few applied problems when a rather complicated mathematical model is used and no a priori information is available. For overcoming the lack of appropriate data, we pose the problem of finding part of the information unknown; namely, we search only for the discontinuity surfaces of the integrand. The corresponding uniqueness theorem is proved. The present paper finalizes our studies into the case of integration over one-dimensional manifolds. In the previous articles we considered similar problems in the case of integration over straight lines. In this paper the same result is proved for the integration of unknown functions over unknown curves.

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“…Further we will continue the investigation of formula (2.1) which was begun in [1]. We plan to prove that the unboundedness of V r /(r,t<;) may be employed in a modification of the gradient method for detection of latent inhomogeneities in a medium.…”

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“…There exists the gradient of the function /(r,o;) for any r G U(r 0 ), ω G Ω, and the following formula takes place Brought to you by | University of Queensland -UQ Library Authenticated Download Date | 6/19/15 2:05 PMwhere /(r, ω) is tiie solution to the boundary value problem (1.1),(1)(2); yj = <j(r, -ω)ω; [ω · V r /(t/j,o;)] is the jump of the expression in square brackets; i.e. lnpr-*w,9God* + const.…”

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A formula for the gradient of the transport equation solution

Аниконов¹

1996

Journal of Inverse and Ill-Posed Problems

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This article is the first stage of the detailed foundation for some results that were partially published [1] and formulated by the author in his report at the international conference IPES-94 (Japan, Osaka, July 27-30, 1994) [2]. The work may be considered as pertaining to the qualitative theory of the transport equation solution. Such investigations, in particular, studying the singularities of the solution, are often used to study inverse problems. Some authors noted the unboundedness of partial derivatives of the transport equation solution and obtained corresponding upper estimates [3][4][5]. However the knowledge of these estimates is not quite enough for inverse problems investigation. One needs also either lower estimates or exact equalities. In this paper the formula for the gradient of the transport equation solution is justified, and its unbounded parts are shown. The application of this formula will be a subject of further author's publications, but some first conclusions are given here.•Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, Radio 7, Vladivostok, 690041, Russia.

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On an inverse problem for the transport equation

Cited by 7 publications

References 4 publications

A Method for Studying Singular Integral Equations

A Method for Studying Singular Integral Equations

An integral geometry underdetermined problem for a family of curves

A formula for the gradient of the transport equation solution

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