Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Poincare Problem for the Stokes-Bitsadze Equation with Supersingular Point in the Junior Coefficients

Author(s):

Yuri Sergeevich Fedorov

Department of Higher Mathematics, Associate Professor
National Research University MPEI
Address: 14 Krasnokazarmennaya St., Moscow, Russia, 111250

fedorovys@mpei.ru

Abstract:

It is proved that any elliptic system of second order equations with constant coefficients with two functions from two variables is reduced to one of the Laplace or Bitsadze equations in complex variables. The Laplace equation has been studied well enough, which cannot be said about the Bitsadze equation. This paper establishes the relationship between the Stokes and Bitsadze equations. In addition, for the Stokes-Bitsadze equation with a supersingular point in the junior coefficients the solution to the Poincare problem in the classes of functions satisfying the Gelder condition is found. After reducing the Poincare problem to the Riemann-Hilbert problem the singularity of the solution of Poincare's problem and representation of the solution as an explicit formula (depending on the index of this problem) are investigated.

Keywords

References:

  1. Bitzadze A. B. Nekotoryye klassy uravneniy v chastnykh proizvodnykh. [ Some Classes of Partial Differential Equations], M. : Nauka, 1981. 448p. (in Russian)
  2. Bochev P. B. Analysis of least-squares finite element methods Muhammad Tahir , A. R. Davies for the Navier-Stokes equations , Siam J. Numer. Anal., 34:5(1997), 1817-1844
  3. M. Tahir, A. R. Davies. Stokes-Bitsadze problem - I . Punjab UniversityJournal of Mathematics (ISSN 1016-2526)Vol. 32(2005) , 77-90
  4. M. Tahir. The Stokes-Bitsadze system, Punjab Univ. J. Math. XXXII (1999) 173-180
  5. T. Vaitekhovich. Boundary value problems to second order complex partial differential equations in a ring domain, Siauliai Math. Semin. 2 :10 (2007), 117-146
  6. Vekua I. N. Obobshchennyye analiticheskiye funktsii. 2-ye izd., [ Generalized analytical functions. 2nd ed. ] , M., Nauka, 1988. 510p. (in Russian)
  7. Muskhelishvili N. I. Singulyarnyye integral'nyye uravneniya [Singular integral equations] M. : Nauka, 1968. 511p. (in Russian)
  8. Soldatov A. P. Krayevaya zadacha lineynogo sopryazheniya teorii funktsiy. Izv. AH SSSR. Sep. mat. [Boundary value problem of linear conjugation of function theory. Izv. Academy of Sciences of the USSR. Ser. mat. ], 43:1( 1979), 184-202. (in Russian)
  9. Abdurauf B. Rasulov ; Yurii S. Fedorov ; Anna M. Sergeeva. Integral Representations of Solutions for the Bitsadze Equation with the set of Supersingular points in the Lower Coefficients, (2019) International Conference on Applied and Engineering Mathematics (ICAEM), IEEEElectronic ISBN: 978-1-7281-2353-0, Print on Demand(PoD) ISBN: 978-1-7281-2354-7 , p. 13 - 17.
  10. B. B. Oshorov. On boundary value problems for the Cauchy-Riemann and Bitsadze systems of equations, Doklady Mathematics , 73:2 (2006) 241-244. (in Russian)
  11. S. Hizliyel, M. Cagliyan. A boundary value problem for Bitsadze equation in matrix form. Turkish J. Math. 35:1 (2011), 29-46

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