Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Salimov R. B. About New Approach to Solution of Riemann’s Boundary Value Problem with Condition on the Half-line in Case of Infinite Index. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 1, pp. 29-33. DOI: 10.18500/1816-9791-2016-16-1-29-33, EDN: VUSODN

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
14.03.2016
Full text:
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Language: 
Russian
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UDC: 
517.54
EDN: 
VUSODN

About New Approach to Solution of Riemann’s Boundary Value Problem with Condition on the Half-line in Case of Infinite Index

Autors: 
Salimov Rasikh Bakhtigareevich, Kazan State University of Architecture and Engineering
Abstract: 

To solve a homogeneous Riemann boundary value problem with infinite index and condition on the half-line we propose a new approach based on the reduction of the considered problem to the corresponding task with the condition on the real axis and finite index. It is required to define a function Φ(z), analytic and bounded in the complex plane z, cut down on positive real semi-axis L+ , if the edge condition Φ+(t) = G(t)Φ−(t), t ∈ L+ is fulfilled, where Φ+(t),     Φ− (t) are limit values of the function Φ(z), as z → tcorrespondingly on the left and on the right,G(t) is a given function, for which argument argG(t) = ν− tρ +ν(t), t ∈ L+  holds, here ν− , ρ are given numbers, ν− > 0, 1/2< ρ < 1, and ln|G(t)|, ν(t) are functions which satisfy the Holder condition. It is admitted that G(t) = 1 at t ∈ (−∞,0). The functions E+ (z) = e(α+iβ)zρ , 0 ≤ argz ≤ π, E− (z) = e(α−iβ)zρ , −π ≤ argz ≤ 0 are used to avoid infinite gap of the argG(t), by the selection of real numbers α, β.

References: 
  1. 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 (in Russian).
  2. Gakhov F. D. Boundary value problems. Moscow, Nauka, 1977, 640 p. (in Russian).
  3. Markushevich A. I. The theory of analytic functions, in 2 vol. Vol. 2. Moscow, Nauka, 1968, 624 p. (in Russian).
  4. Govorov N. V. Riemann’s boundary problem with infinite index. Moscow, Nauka, 1986, 239 p. (in Russian).
Received: 
17.11.2015
Accepted: 
25.02.2016
Published: 
31.03.2016