In the paper in a class of regular harmonic functions we study properties of some integro-differential operators that generalize the operators of fractional differentiation in Hadamard sense. These operators transfer regular harmonic functions to the same function, and are inverse to the regular harmonic functions. Boundary value problem with the boundary operator of fractional order is studied in the exterior of the unit sphere. The considered problem generalizes the well-known Neumann problem on boundary operators of fractional order. We prove a theorem on existence and uniqueness of solutions of the problem. Moreover, an integral representation of the problem solution is obtained.
REFERENCES
1.
Bitsazde
, A. V.
, Equations of Mathematical Physics
, Mir Publishers
, Moscow
, 1980
. (Russian)2.
3.
Kilbas
, A.A.
, Srivastava
, H.M.
, Trujillo
, J.J.
, Theory and Applications of Fractional Differential Equations
, Elsevier
. North-Holland
. Mathematics studies. 2006
.4.
Berdyshev
, A.S.
, Cabada
, A.
, Turmetov
, B. Kh.
, ”On solvability of a boundary value problem for a non-homogeneous biharmonic equation with a boundary operator of a fractional order
”. Acta Mathematica Scientia
. 34
(6
), 1695
–1706
(2014
).5.
Karachik
, V.V.
, Turmetov
, B.Kh.
, Torebek
, B.T.
, ”On some integro-differential operators in the class of harmonic functions and their applications
”. Siberian Advances in Mathematics
. 22
(2
), 115
–134
(2012
).6.
Kirane
, M.
, Torebek
, B.T.
, ”On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions
”. Mathematical Methods in the Applied Sciences
(2015
)7.
Krasnoschok
, M.
, Vasylyeva
, N.
, ”On a nonclassical fractional boundary-value problem for the Laplace operator
”. Journal of Differential Equations
. 257
, 1814
–1839
(2014
).8.
Muratbekova
, M. A.
, Shynaliev
, K.M.
, Turmetov
, B. Kh.
, ” On solvability of a nonlocal problem for the Laplace equation with the fractional-order boundary operator
”. Boundary Value Problems
. 2014
(29
),(2014
).9.
Sadybekov
, M.A.
, Turmetov
, B.Kh.
, Torebek
, B.T.
, ” Solvability of nonlocal boundary-value problems for the Laplace equation in the ball
”. Electronic Journal of Differential Equations
. 2014
(157
), 1
–14
(2014
).10.
Torebek
, B.T.
, Turmetov
, B.Kh.
, ” On solvability of a boundary value problem for the Poisson equation with the boundary operator of a fractional order
”. Boundary Value Problems
. 2013
(93
), (2013
)11.
Turmetov
, B.Kh.
, Torebek
, B.T.
, ”Modified Bavrin operators and their applications
” Differential Equations
.51
(2
), 243
–254
(2015
).12.
Turmetov
, B.Kh.
, Koshanova
, M.D.
, Usmanov
, K.I.
, ”Solvability of boundary-value problems for Poisson equations with Hadamard type boundary operator
”. Electronic Journal of Differential Equations
. 2016
(161
), 1
–12
(2016
).13.
Umarov
, S.R.
, ”On some boundary value problems for elliptic equations with boundary operator of fractional order
”. Doklady Mathematics
. 48
(3
), 655
–658
(1994
).14.
Umarov
, S.R.
, Luchko
, Y.
, Gorenflo
, R.
, ”On boundary value problems equations with boundary operators of fractional order
”. Fractional Calculus and Applied Analysis
. 3
(4
), 454
–468
(2000
).15.
Turmetov
, B.Kh.
, Torebek
, B. T.
, ”On solvability of exterior boundary value problem with fractional boundary condition
”. International Conference on Analysis And Applied Mathematics
. AIP Conference Proceedings
. 1676
(020096
) (2015
).16.
Torebek
, B. T.
, Turmetov
, B.Kh.
, ”Questions on solvability of exterior boundary value problem with fractional boundary conditions
”. Electronic Journal of Qualitative Theory of Differential Equations
. 2016
(25
), 1
–14
(2016
).17.
Akhmedov
, T.M.
, Veliev
, E.
, Ivakhnychenko
, M.
, ”Fractional operators approach in electromagnetic wave reflection problems
”. Journal of electromagnetic waves and applications
. 21
(13
), 1787
–1802
(2007
).18.
Akhmedov
, T.M.
, Veliev
, E.
, Ivakhnychenko
, M.
, ” Description of the boundaries in scattering problems using fractional operators
”. Radio Physics and Electronics
.14
, 133
–141
(2009
).19.
Veliev
, E.
, Ivakhnychenko
, M.
, ” Fractional boundary conditions in plane wave diffraction on a strip
”. Progress In Electromagnetics Research
. 79
, 443
–462
(2008
).20.
Lifanov
, I. K.
, ”A singular integral-equation of the first kind for a Neumann problem
”. Differential Equations
. 24
(1
), 87
–91
(1988
).
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