Riemann boundary value problem

The Solution of the Homogeneous Boundary Value Problem of Riemann with Infinite Index of Logarithmic Order on the Beam by a New Method

Authors: Salimov R. B., Khasanova E. N.

In this paper we consider the homogeneous Riemann boundary value problem with infinite index of logarithmic order and boundary condition on the unlimited ray. This ray goes by the positive real axis and has a vertex. We solve the problem for analytic function with the cut along the ray. The value of the function at any point of the left bank equals the product of the coefficient and the value of the function at the corresponding point of the right bank of the cut. Let the modulus of the coefficient  meet the Holder condition at each point of the ray.

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

Authors: Salimov R. B.

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.

The Solution of the Homogeneous Riemann Boundary Value Problem with a Countable Set of Points of Discontinuity of the First Kind its Coefficient

Authors: Salimov R. B.

We consider the Riemann homogeneous boundary value problem with a countable set of points of discontinuity of the first kind in the case, when it is required to find two functions, analytic, respectively, in the upper and lower half-plane, for a given linear boundary condition on the real axis, connecting the boundary values of the unknown functions.

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

Authors: Salimov R. B., Karabasheva E. N.

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.