Inversion of integral of B-potential type with density from Φγ

Shishkina, Elina

doi:10.1007/s10958-009-9487-y

Cited by 11 publications

(5 citation statements)

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“…In works of Muhlisov and his disciples, problems for B-potentials are considered. In works of Shishkina, Aisgersson-type mean-value theorems and new problems for the Euler-Poisson-Darboux equation are investigated, new classes of Riesz-type potentials with B-hyperbolic and ultrahyperbolic operators are constructed, and applications of these results to differential equations of the corresponding types are considered (see [405][406][407][408][616][617][618][619][620][621][622]). The applying of transmutation operators and related methods in the theory of inverse problems and scattering theory is continued (see [38,239,291,414,453,596]).…”

Section: Theory Of Transmutation Operators: History and Contemporary ...mentioning

confidence: 99%

The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

Katrakhov¹,

Вячеславович²,

Sitnik

et al. 2018

SMFN

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The main content of this book is composed from two doctoral theses: by V. V. Katrakhov (1989) and by S. M. Sitnik (2016). In our work, for the ﬁrst time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to diﬀerential equations with singularities in coeﬃcients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Signiﬁcant part of these results is published with detailed proofs for the ﬁrst time. In the ﬁrst chapter, we give historical background, necessary notation, deﬁnitions, and auxiliary facts. In the second chapter, we give the detailed theory of Sonin and Poisson transmutations. In the third chapter, we describe an important special class of the Buschman-Erde´lyi transmutations and their applications. In the fourth chapter, we consider new weighted boundary-value problems with Sonin and Poisson transmutations. In the ﬁfth chapter, we consider applications of the Buschman-Erde´lyi transmutations of special form to new boundary-value problems for elliptic equations with signiﬁcant singularities of solutions. In the sixth chapter, we describe a universal compositional method for construction of transmutations and its applications. In the concluding seventh chapter, we consider applications of the theory of transmutations to diﬀerential equations with variable coeﬃcients: namely, to the problem of construction of a new class of transmutations with sharp estimates of kernels for perturbed diﬀerential equations with the Bessel operator, and to special cases of the well-known Landis problem on exponential estimates of the rate of growth for solutions of the stationary Schro¨dinger equation. The book is concluded with a brief biographic essay about Valeriy V. Katrakhov, as well as detailed bibliography containing 648 references.

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Section: Theory Of Transmutation Operators: History and Contemporary ...mentioning

confidence: 99%

The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

Katrakhov¹,

Вячеславович²,

Sitnik

et al. 2018

SMFN

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“…Such operators in the case of negative powers are analogues of the Riesz potentials with Euclidean distance, and for them the term elliptic B-potentials is accepted. Elliptic B-potentials and its inverses have been studied in papers [12]- [26] and this list is not complete. It is interesting that the Bessel operator and the hyper-Bessel operator could be considered as generalized fractional derivatives too (see [27], p. 97).…”

Section: Brief Historymentioning

confidence: 99%

Inversion of the Mixed Riesz Hyperbolic B-Potentials

Shishkina¹

2017

Int. J. of Appl. Math.

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“…[15][16][17][18][19][20] Riesz B-potential with Euclidean distance in the weighted Lebegue spaces and its inverse are given in references herein. [21][22][23][24][25][26][27] In Sarikaya et al, 28 for the operator similar to operator (3), some properties other than given in this article were obtained. The operator (3) and the corresponding integral kernels were studied in recent works.…”

Section: Introductionmentioning

confidence: 97%

“…Sobolev theorems for potential generated by Bessel operator with Euclidean distance in the Morrey‐Bessel and BMO‐Bessel spaces were considered in previous studies . Riesz B‐potential with Euclidean distance in the weighted Lebegue spaces and its inverse are given in references herein . In Sarikaya et al, for the operator similar to operator , some properties other than given in this article were obtained.…”

Section: Introductionmentioning

confidence: 99%

Weighted generalized functions and hyperbolic Riesz B‐potential

Shishkina

2019

Math Methods in App Sciences

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The paper is devoted to the study of the fractional integral operator, which is a negative real power of the singular wave operator generated by Bessel operator using weighted generalized functions. We give conditions for this operator to be bounded in appropriate spaces, obtain formula for the Hankel transform of this operator, and get formula of connection between this operator and natural degree of singular wave operator generated by Bessel operator.

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Inversion of integral of B-potential type with density from Φγ

Cited by 11 publications

References 1 publication

The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

Inversion of the Mixed Riesz Hyperbolic B-Potentials

Weighted generalized functions and hyperbolic Riesz B‐potential

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