Cite this article as:

Salimov R. B., Karabasheva E. N. The new approach to solving the Riemann boundary value problem with infinite index . Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 2, pp. 155-165. DOI: https://doi.org/10.18500/1816-9791-2014-14-2-155-165

Language: 
Russian
Heading: 
Mathematics
UDC: 
517.54

The new approach to solving the Riemann boundary value problem with infinite index

Authors: 
Salimov Rasikh Bakhtigareevich, Kazan State University of Architecture and Engineering, Kazan State University of Architecture and Engineering, Russia, 420043, Kazan, Zelenaya st., 1 , salimov.rsb@gmail.com
Karabasheva Enge Nazipovna, Kazan State University of Architecture and Engineering, 1, Zelenaya str., 420043, Kazan, Tatarstan, Russia , enkarabasheva@bk.ru
Abstract: 

This research considers Riemann–Hilbert boundary value problem with infinite index where edge condition of problem is established by the real axis. To solve this problem the approach based on the removal of the infinite discontinuity of the argument of boundary condition coefficient is used. The approach is analogous to the one which, in the context of the finite index of the problem in researches by F. D. Gakhov, helps to remove a discontinuity of initial genre of boundary condition coefficient with specially created functions, different from the ones in this research.

Key words: 
Riemann boundary value problem, analytical function, infinite index.
DOI: 
https://doi.org/10.18500/1816-9791-2014-14-2-155-165
References
1. Muskhelishvili N. I. Singular integral equations. Moscow, Nauka, 1968, 511 p. (in Russian).
2. Gakhov F. D. Boundary value problems. Moscow, Nauka, 1977, 640 p. (in Russian).
3. Govorov N. V. Riemann’s boundary problem with infinite index. Moscow, Nauka, 1986, 239 p. (in Russian).
4. Tolochko M. E. About the solvability of the homogeneous Riemann boundary value problem for the half-plane with infinite index. Izvestiya Akad. Nauk BSSR. Ser. Fiz.-mat. nauki, 1972, no. 5, pp. 34–41 (in Russian).
5. Sandrygailo I. E. On Hilbert –Rieman boundary value problem for half-plane with infinite index. Izvestiya Akad. Nauk BSSR. Ser. Fiz.-Mat. Nauki, 1974, no. 6, pp. 872–875 (in Russian).
6. Monahov V. N., Semenko E. V. Boundary value problem with infinite index in Hardy spaces. Dokl. Akad. Nauk SSSR, 1986, vol. 291, no. 3, pp. 544–547 (in Russian).
7. Alehno A. G. Sufficient conditions for the solvability of homogeneous Riemann boundary value problem with infinite index. Trudy matematicheskogo tsentra imeni N. I. Lobachevskogo, Kazan, 2002, vol. 14, pp. 71–77 (in Russian).
8. Garifianov F. N. About a special case of the Riemann problem. Trydu seminara po kraevum zadacham. Kazan, 1984, no 22, pp. 66–68 (in Russian).
9. Katc B. A. About Riemann problem with an oscillating coefficient. Trydu seminara po kraevum zadacham. Kazan, 1977, no. 14, pp. 110–120 (in Russian).
10. Salimov R. B., Shabalin P. L. The regularizing factor method for solving a homogeneous Hilbert problem with an infinite index. Russian Math. [Izvestiya VUZ. Matematika], 2001, vol. 45, iss. 4, pp. 74–77.
11. Markushevich A. I. The theory of analytic functions : in 2 vol. Moscow, Nauka, 1968, vol. 2, 624 p. (in Russian).
12. Levin B. Ya. Distribution of zeros of entire functions. Moscow, Gostechizdat, 1956, 632 p. (in Russian).
13. Salimov R. B., Shabalin P. L. Solution of the Hilbert Problem with infinite index. Math. Notes, 2003, vol. 73, iss. 5, pp. 680–689. DOI: 10.4213/mzm221.
Short text (in English): 
page_05.pdf
58
Full text:
download
77