Geodesic
The Riemann problem in a half-plane for generalized analytic functions with a singular line
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2023), pp. 78-89.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we have studied an inhomogeneous boundary value problem Riemann with a finite index and a boundary condition on the real axis for one generalized Cauchy-Riemann equation with singular coefficients. To solve this problem, we derived a structural formula for the general solution of the generalized equation and conducted a complete study of the solvability of the Riemann boundary value problem of the theory of analytic functions with an infinite index of logarithmic order. Based on the results of this study, we derived a formula for a general solution and studied the existence and number of solutions boundary value problems for generalized analytical functions.
Keywords: Riemann problem, generalized analytic function, infinite index, entire function of given order zero. 
@article{IVM_2023_3_a6,
     author = {P. L. Shabalin and R. R. Faizov},
     title = {The {Riemann} problem in a half-plane for generalized analytic functions with a singular line},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {78--89},
     publisher = {mathdoc},
     number = {3},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2023_3_a6/}
}
TY  - JOUR
AU  - P. L. Shabalin
AU  - R. R. Faizov
TI  - The Riemann problem in a half-plane for generalized analytic functions with a singular line
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2023
SP  - 78
EP  - 89
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2023_3_a6/
LA  - ru
ID  - IVM_2023_3_a6
ER  - 
%0 Journal Article
%A P. L. Shabalin
%A R. R. Faizov
%T The Riemann problem in a half-plane for generalized analytic functions with a singular line
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2023
%P 78-89
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2023_3_a6/
%G ru
%F IVM_2023_3_a6
P. L. Shabalin; R. R. Faizov. The Riemann problem in a half-plane for generalized analytic functions with a singular line. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2023), pp. 78-89. http://geodesic.mathdoc.fr/item/IVM_2023_3_a6/

[1] Mikhailov L.G., Novye klassy osobykh integralnykh uravnenii i ikh primenenie k differentsialnym uravneniyam s singulyarnymi koeffitsientami, Izd-vo AN Tadzh. SSR, Dushanbe, 1963 | MR

[2] Radzhabov N.R., Integralnye predstavlenie i granichnye zadachi dlya nekotorykh differentsialnykh uravnenii s singulyarnoi liniei ili singulyarnymi poverkhnostyami, v. 1, Izd-vo TGU, Dushanbe, 1980

[3] Radzhabov N.R., “Integralnye predstavleniya i granichnye zadachi dlya obobschennoi sistemy Koshi-Rimana s singulyarnoi liniei”, Dokl. AN SSSR, 267:2 (1982), 300–305 | MR | Zbl

[4] Radzhabov N.R., Rasulov A. B., “Integralnye predstavlenie i granichnye zadachi dlya odnogo klassa sistem differentsialnykh uravnenii ellipticheskogo tipa s singulyarnym mnogoobraziem”, Differents. uravneniya, 25:7 (1989), 1279–1981 | MR

[5] Usmanov Z.D., Obobschennye sistemy Koshi–Rimana s singulyarnoi tochkoi, Izd-vo AN Tadzh. SSR, Dushanbe, 1993

[6] Begehr H., Dao-Qing Dai., “On continuous solutions of a generalized Cauchi–Riemann system with more than one singularity”, J. Diff. Equat., 196 (2004), 67–90 | DOI | MR | Zbl

[7] Meziani A., “Representation of solutions of a singular CR equation in the plane”, Complex Var. and Elliptic. Eguat., 53 (2008), 1111–1130 | DOI | MR | Zbl

[8] Rasulov A.B., Soldatov A.P., “Kraevaya zadacha dlya obobschennogo uravneniya Koshi–Rimana s singulyarnymi koeffitsientami”, Differents. uravneniya, 52:5 (2016), 637–650 | DOI | Zbl

[9] Fedorov Yu.S., Rasulov A.B., “Zadachi tipa Gilberta dlya uravneniya Koshi–Rimana s singulyarnymi okruzhnostyu i tochkoi v mladshikh koeffitsientakh”, Differents. uravneniya, 57:1 (2021), 140–144 | DOI | Zbl

[10] Rasulov A.B., “Zadacha Rimana na poluokruzhnosti dlya obobschennoi sistemy Koshi–Rimana s singulyarnoi liniei”, Differents. uravneniya, 40:9 (2004), 1290–1292 | MR | Zbl

[11] Rasulov A.B., “Integralnye predstavleniya i zadacha lineinogo sopryazheniya dlya obobschennoi sistemy Koshi–Rimana s singulyarnym mnogoobraziem”, Differents. uravneniya, 36:2 (2000), 270–275 | MR | Zbl

[12] Govorov N.V., Kraevaya zadacha Rimana s beskonechnym indeksom, Nauka, M., 1986

[13] Monakhov V.N., Semenko E.V., Kraevye zadachi i psevdodifferentsialnye operatory na rimanovykh poverkhnostyakh, Fizmatlit, M., 2003

[14] Ostrovskii I.V., “Odnorodnaya kraevaya zadacha Rimana s beskonechnym indeksom na krivolineinom konture I”, Teoriya funktsii, funkts. analiz i ikh prilozheniya, 1991, no. 56, 95–105

[15] Yurov P.G., “Neodnorodnaya kraevaya zadacha Rimana s beskonechnym indeksom logarifmicheskogo poryadka $\alpha\geq 1$”, Mater. Vsesoyuzn. konferentsii po kraevym zadacham, Kazan, 1970, 279–284 | Zbl

[16] Alekna P.Yu., “Neodnorodnaya kraevaya zadacha Rimana s beskonechnym indeksom logarifmicheskogo poryadka dlya poluploskosti”, Litovsk. matem. sb., 1974, no. 3, 5–18 | MR | Zbl

[17] Gakhov F.D., Kraevye zadachi, Nauka, M., 1977

[18] Melnik I.M., “O kraevoi zadache Rimana s razryvnymi koeffitsientami”, Izv. vuzov. Matem., 1959, no. 2, 158–166 | Zbl

[19] Vekua I.N., Obobschennye analiticheskie funktsii, Nauka, M., 1988

[20] Salimov R.B., Karabasheva E.N., “Novyi podkhod k resheniyu kraevoi zadachi Rimana s beskonechnym indeksom”, Izv. Saratovsk. un-ta. Nov. ser. Ser.: Matem. Mekhan. Informatika, 14:2 (2014), 155–165 | Zbl

[21] Alekhno A.G., “Kraevaya zadacha Gilberta s beskonechnym indeksom logarifmicheskogo poryadka”, Dokl. NAN Belarusi, 53:2 (2009), 5–10 | MR | Zbl