#### For citation:

Perelman N. R. On One Exceptional Case of the First Basic Three-Element Carleman-Type Boundary Value Problem for Bianalytic Functions in a Circle. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2020, vol. 20, iss. 2, pp. 185-192. DOI: 10.18500/1816-9791-2020-20-2-185-192, EDN: UTHKAU

# On One Exceptional Case of the First Basic Three-Element Carleman-Type Boundary Value Problem for Bianalytic Functions in a Circle

This article considers a non-degenerate (nonreducible to two-element) three-element problem of Carleman type for bianalytic functions in an exceptional case, that is, when one of the coefficients of the boundary condition vanishes at a finite number of contour points. The unit circle is taken as the contour. For this case, an algorithm for solving the problem is constructed, which consists in reducing the boundary conditions of this problem to a system of four Fredholm type equations of the second kind. For this, the boundary value problem for bianalytic functions is represented as two boundary value problems of Carleman type in the class of analytic functions, then, by introducing auxiliary functions, these problems are represented as scalar Riemann problems in the exceptional case. Using the well-known formulas for solving such problems, we reduce each of the boundary conditions of Carleman-type problems for analytic functions to a pair of well-studied equations of the Fredholm type of the second kind.

- Rasulov K. M. Kraevye zadachi dlya polianaliticheskikh funktsiy i nekotorye ikh prilozheniya [Boundary-value problems for polyanalytic functions and some of their applications]. Smolensk, Izd-vo Smolenskogo gosudarstvennogo pedagogicheskogo universiteta, 1998. 345 p. (in Russian).
- Perelman N. R., Rasulov K. M. Three-element problem of Carleman type for bianalitic functions in a circle. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2012, vol. 12, iss. 2, pp. 18–26 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2012-12-2-18-26
- Rasulov K. M., Titov O. A. On the study of the first basic three-element Carleman type boundary value problem for bianalytic functions in a circle. Sistemy komp’yuternoy matematiki i ikh prilozheniya [Computer mathematics systems and their applications: Proc. Int. Conf.]. Smolensk, Izd-vo Smolenskogo gosudarstvennogo universiteta, 2005, iss. 6, pp. 148–154 (in Russian).
- Perelman N. R. On the solution of the first three-element Carleman type problem for bianalytic functions in the nondegenerate case. Nekotorye aktual’nye problemy sovremennoy matematiki i matematicheskogo obrazovaniya: materialy mezhdunar. konf. “Gertsenovskie chteniya–2011”. LXIV [Some Pressing Problems of Modern Mathematics and Mathematical Education: Proc. Int. Conf. “Herzen Readings–2011”. LXIV]. St. Petersburg, BAN Publ., 2011, pp. 152–156 (in Russian).
- Litvinchuk G. S. Solvability theory of boundary value problems and singular integral equations with shift. Dordrecht, Boston, Kluwer Academic Publ., 2000. 378 p. DOI: https://doi.org/10.1007/978-94-011-4363-9 (Russ. ed.: Moscow, Nauka, 1977. 448 p.).
- Rasulov K. M. The integral trap method for solving a three-element boundary value problem with Carleman shift in classes of analytic functions. Sistemy komp’yuternoy matematiki i ikh prilozheniya [Computer mathematics systems and their applications: Proc. Int. Conf.]. Smolensk, Izd-vo Smolenskogo gosudarstvennogo universiteta, 2012, iss. 2, pp. 191–212 (in Russian).
- Rasulov K. M. Metod sopriazheniya analiticheskikh funktsiy i nekotorye ego prilozheniya [Conjugation method of analytic functions and some of its applications]. Smolensk, Izd-vo Smolenskogo gosudarstvennogo universiteta, 2013. 189 p. (in Russian).
- Gakhov F. D. Kraevye zadachi [Boundary value problems]. Moscow, Nauka, 1977. 640 p. (in Russian).
- Chikin L. A. Special cases of the Riemann boundary value problem and singular integral equations. Uchenye zapiski Kazanskogo universiteta [Scientific Notes of Kazan University], 1953, vol. 113, no. 10, pp. 57–105 (in Russian).
- Perelman N. R., Rasulov K. M. Three-Element One-Sided Boundary Value Problem for Analytic Functions with a Reverse Shift of Carleman in Exceptional Case. Izv. Brjanskogo Gos. Univ. [The Bryansk State University Herald], 2012, no. 4 (2), pp. 44–51 (in Russian).
- Markushevich A. I. Teoriya analiticheskikh funktsiy [The theory of analytic functions: in 2 vols.]. Moscow, Nauka, 1967–1968. Vol. 1. 488 p. (in Russian).
- Mushelishvili N. I. Singulyarnye integral’nye uravneniya [Singular integral equations]. Moscow, Nauka, 1968. 513 p. (in Russian).
- Rasulov K. M. Three-element one-sided boundary value problem with Carleman shift in classes of analytic functions in a circle. Izvestiya SmolGU, 2008, no. 2, pp. 94–104 (in Russian).
- Rasulov K. M. On the solution of a three-element one-sided boundary value problem with a Carleman shift in classes of analytic functions in a circle. Vesnik Hrodzienskaha dziaranaha univiersiteta imia Janki Kupaly. Ser. 2. Matematyka. Fizika. Infarmatyka, 2010, no. 3 (102), pp. 31–37 (in Russian).
- Rasulov K. M. On the solution of a three-element boundary value problem with a Carleman shift for analytical functions in the nondegenerate case. Vestnik Yuzhno-Ural’skogo Gosudarstvennogo Universiteta. Ser. Matematika. Mekhanika. Fizika, 2012, no. 34, pp. 43–52 (in Russian).

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