Representation of the General Solution of Cauchy-Riemann Type Equation with Singular Circumference and a Singular Point
Author(s):
Abdurauf Babadzhanovich Rasulov
FGBOU VO "NIU "MPEI"
(Federal state budget educational institution of
higher professional education "national research University "MPEI"),
associate Professor of the Department of mathematics,
PhD in physics and mathematics, associate Professor
rasulov_abdu@rambler.ru
Maushkura A. Bobodzhanova
Department of mathematics,
National research University MPEI,
Moscow
bobojanova@mpei.ru
Yury Sergeevich Fedorov
FGBOU VO "NIU "MPEI"
(Federal state budget educational institution of
higher professional education "national research University "MPEI"),
associate Professor of the Department of mathematics
Abstract:
In the theory of differential equations in partial derivatives
the systems of the Cauchy-Riemann equations
with regular and singular coefficients occupy a higly important place.
The theory of such equations
with regular coefficients was investigated deeply enough. It is not so for
the systems of Cauchy-Riemann type with singular coefficients.
The applications of such systems in mapy tasks attracts the attention of researches
to the theory.
Note that in works of many authors solutions of the Cauchy-Riemann
system with a singular point were found in the form of a series, and the compactness
of the main integral operator was proved only in a small neighborhood
of a singular point or on "smallness" conditions on the coefficients.
Previously differential equations with a singular point and a singular line
were studied separately.
So far the obtaining integral representations of the general solution of equations
with the Cauchy-Riemann operator with singularities in the coefficients
for different varieties is little studied,
although there are many examples confirming the importance of the
application of such equations.
In this connection differential
equations with a singular point and segments or more complex singular
manifolds (for example a circle) are the object of our research.
In this paper we consider the generalized system of Cauchy-Riemann
with complex conjugation, whose coefficients have singularities on the circle
and in a point. On the basis of the constructed
resolvent we found an integral representation of the general solution.
In all these cases a special part of a solution is separated
that allows us to study the behavior of solutions in a
neighborhood of singular manifolds in detail. Thus the
integral representation of the general solution may be applied to the study of
boundary value problems.
Keywords
- generalized system of Cauchy-Riemann type
- singular integral equation
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