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CMFD, 2018, Volume 64, Issue 2, Pages 211–426 (Mi cmfd355)  

This article is cited in 5 scientific papers (total in 5 papers)

The transmutation method and boundary-value problems for singular elliptic equations

V. V. Katrakhova, S. M. Sitnikb

a Voronezh and Vladivostok, Russia
b Belgorod National Research University "Belgorod State University", Institute of Engineering Technology and Natural Science, Belgorod, Russia

Abstract: 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–Erdéelyi 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–Erdélyi 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 Schrödinger 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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UDC: 517.956.22

Citation: V. V. Katrakhov, S. M. Sitnik, “The transmutation method and boundary-value problems for singular elliptic equations”, Singular differential equations, CMFD, 64, no. 2, Peoples' Friendship University of Russia, M., 2018, 211–426

Citation in format AMSBIB
\by V.~V.~Katrakhov, S.~M.~Sitnik
\paper The transmutation method and boundary-value problems for singular elliptic equations
\inbook Singular differential equations
\serial CMFD
\yr 2018
\vol 64
\issue 2
\pages 211--426
\publ Peoples' Friendship University of Russia
\publaddr M.

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    1. F. G. Khushtova, “K probleme edinstvennosti resheniya zadachi Koshi dlya uravneniya drobnoi diffuzii s operatorom Besselya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 22:4 (2018), 774–784  mathnet  crossref  elib
    2. E. L. Shishkina, “Obschee uravnenie Eilera—Puassona—Darbu i giperbolicheskie $B$-potentsialy”, Uravneniya v chastnykh proizvodnykh, SMFN, 65, no. 2, Rossiiskii universitet druzhby narodov, M., 2019, 157–338  mathnet  crossref
    3. K. B. Sabitov, N. V. Zaitseva, “Vtoraya nachalno-granichnaya zadacha dlya $B$-giperbolicheskogo uravneniya”, Izv. vuzov. Matem., 2019, no. 10, 75–86  mathnet  crossref
    4. M. V. Polovinkina, I. P. Polovinkin, “O svoistvakh rimanovykh metrik, svyazannykh s $B$-ellipticheskimi operatorami”, Materialy Voronezhskoi zimneimatematicheskoi shkolySovremennye metody teorii funktsiii smezhnye problemy.28 yanvarya–2 fevralya 2019 g.Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 170, VINITI RAN, M., 2019, 31–37  mathnet  crossref  elib
    5. A. V. Glushak, “Operatornye gipergeometricheskie funktsii”, Geometriya i mekhanika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 174, VINITI RAN, M., 2020, 37–45  mathnet  crossref
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