Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Shabalin P. L., Faizov R. R. The Riemann problem on a ray for generalized analytic functions with a singular line. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 1, pp. 58-69. DOI: 10.18500/1816-9791-2023-23-1-58-69, EDN: UYQLJS

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.03.2023
Full text:
(downloads: 983)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
517.54
EDN: 
UYQLJS

The Riemann problem on a ray for generalized analytic functions with a singular line

Autors: 
Shabalin Pavel Leonidovich, Kazan State University of Architecture and Engineering
Faizov Rafael Rustamovich, Kazan State University
Abstract: 

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on a ray for a generalized Cauchy – Riemann equation with a singular coefficient. For the solution of this problem, we derived a formula for the general solution of the generalized Cauchy – Riemann equation under constraints that led to an infinite index of logarithmic order of the accompanying problem for analytical functions. We have obtained a formula for the general solution of the Riemann problem and conducted a complete study of the existence and the number of solutions of a boundary value problem for generalized analytic functions with a singular line.

References: 
  1. Vekua I. N. Obobshchennye analiticheskie funktsii [Generalized Analytic Functions]. Moscow, Nauka, 1988. 507 p. (in Russian).
  2. Mikhailov L. G. Novye klassy osobykh integral’nykh uravneniy i ikh primenenie k differentsial’nym uravneniyam s singulyarnymi koeffitsiyentami [New Classes of Singular Integral Equations and Their Application to Differential Equations with Singular Coefficients]. Dushanbe, TadjikNIINTI, 1963. 183 p. (in Russian).
  3. Radzhabov N. R. Integralnye predstavleniya i granichnye zadachi dlya nekotorykh differentsial’nykh uravneniy s singuliarnoy liniey ili singuliarnymi poverkhnostyami [Integral Representations and Boundary Value Problems for Some Differential Equations with Singular Line or Singular Surfaces]. Vol. 1. Dushambe, Tadzhikskiy gosuniversitet Publ., 1980. 147 p. (in Russian).
  4. Radzhabov N. R. Integral’nye predstavleniya i granichnye zadachi dlya nekotorykh differentsial’nykh uravneniy s singuliarnoi liniei ili s singuliarnymi poverkhnostiami [Integral Representations and Boundary Value Problems for Some Differential Equations with Singular Line or Singular Surfaces]. Vol. 2. Dushambe, Tadzhikskiy gosuniversitet Publ., 1981. 170 p. (in Russian).
  5. Radzhabov N. R. Integral representations and boundary value problems for the generalized Cauchy – Riemann system with a singular line. Doklady Akademii Nauk SSSR, 1982, vol. 267, iss. 2, pp. 300–305 (in Russian). Available at: https://mi.mathnet.ru/dan45725 (accessed 2 August 2022).
  6. Radzhabov N. R., Rasulov A. B. Integral representations and boundary value problems for a class of systems of differential equations of elliptic type with singular manifolds. Differentsial’nye Uravneniya, 1989, vol. 25, iss. 7, pp. 1279–1981 (in Russian). Available at: https://mi.mathnet.ru/de6927 (accessed 2 August 2022).
  7. Usmanov Z. D. Generalized Cauchy – Riemann Systems with a Singular Point. New York, Routledge, 1997. 232 p. (Russ. ed.: Dushambe, TadzhikNIINTI, 1993. 245 p.). https://doi.org/10.1201/9780203753750
  8. Begehr H., Dao-Qing Dai. On continuous solutions of a generalized Cauchi – Riemann system with more than one singularity. Journal of Differential Equations, 2004, vol. 196, iss. 1, pp. 67–90. https://doi.org/10.1016/j.jde.2003.07.013
  9. Meziani A. Representation of solutions of a singular CR equation in the plane. Complex Variables and Elliptic Equations, 2008, vol. 53, iss. 12, pp. 1111–1130. Available at: https://doi.org/ 10.1080/17476930802509239 (accessed 2 August 2022).
  10. Rasulov A. B. Representation of the variety of solutions and investigation of the boundary value problems for some generalized Cauchy – Riemann systems with one and two singular lines. Izvestiya AN Tadzh. SSR. Seriia fiziko-matematicheskikh, khimicheskikh i geologicheskikh nauk, 1982, iss. 4 (84), pp. 23–32 (in Russian).
  11. Rasulov A. B., Soldatov A. P. Boundary value problem for a generalized Cauchy – Riemann equation with singular coefficients. Differential Equations, 2016, vol. 52, iss. 5, pp. 616–629. https://doi.org/10.1134/S0374064116050083
  12. Fedorov Iu. S., Rasulov A. B. Hilbert type problem for a Cauchy – Riemann equation with singularities on a circle and at a point in the lower-order coefficients. Differential Equations, 2021, vol. 57, iss. 1, pp. 127–131. https://doi.org/10.1134/S0012266121010122
  13. Rasulov A. B. The Riemann problem on a semicircle for a generalized Cauchy – Riemann system with a singular line. Differential Equations, 2004, vol. 40, iss. 9, pp. 1364–1366. https://doi.org/10.1007/s10625-005-0015-7
  14. Rasulov A. B. Integral representations and the linear conjugation problem for a generalized cauchy-riemann system with a singular manifold. Differential Equations, 2000, vol. 36, iss. 2, pp. 306–312. https://doi.org/10.1007/BF02754217
  15. Govorov N. V. Riemann’s Boundary Problem with Infinite Index. Operator Theory: Advances and Applications, vol. 67. Berlin, Birkhauser Basel, 1994. 263 p. https://doi.org/10.1007/978-3-0348-8506-5 (Russ. ed.: Moscow, Nauka, 1986. 240 p.).
  16. Monakhov V. N., Semenko E. V. Kraevye zadachi i psevdodifferentsial’nye operatory na rimanovykh poverkhnostyakh [Boundary Value Problems and Pseudodifferential Operators on Riemann Surfaces]. Moscow, Fizmatlit, 2003. 416 p. (in Russian). EDN: UGLDLN
  17. Ostrovskii I. V. Homogeneous Riemann boundary value problem with infinite index on a curved contour. Theory of Functions, Functional Analysis and Their Applications, 1991, iss. 56, pp. 95–105 (in Russian).
  18. Yurov P. G. Inhomogeneous Riemann boundary value problem with infinite index of logarithmic order a > 1. In: Materialy Vsesoyuznoy konferentsii po kraevym zadacham [Materials of the All-Union Conference on Boundary Value Problems]. Kazan, Kazan State University Publ., 1970, pp. 279–284 (in Russian).
  19. Yurov P. G. The homogeneous Riemann boundary value problem with an infinite index of logarithmic type. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1966, iss. 2, pp. 158–163 (in Russian). Available at: https://mi.mathnet.ru/ivm2700 (accessed 2 August 2022).
  20. Salimov R. B., Khasanova E. N. The solution of the homogeneous boundary value problem of Riemann with infinite index of logarithmic order on the beam by a new method. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 2, pp. 160–171 (in Russian). https://doi.org/10.18500/1816-9791-2017-17-2-160-171
Received: 
09.08.2022
Accepted: 
26.09.2022
Published: 
01.03.2023