T-Comm_Article 8_9_2020

STABILITY OF SYSTEMS OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL

DOI: 10.36724/2072-8735-2020-14-9-48-55

Aleksey V. Yudenkov, Smolensk State Academy of Physical Culture, Sport and Tourism, Smolensk, Russia, aleks-ydenkov@mail.ru
Aleksandr M. Volodchenkov, Smolensk Branch of Plekhanov Russian University of Economics;
Smolensk Branch of Saratov State Law Academy, Smolensk, Russia, alexmw2012@yandex.ru
Liliya P. Rimskaya, Smolensk Branch of Plekhanov Russian University of Economics, Smolensk, Russia, aleks-ydenkov@mail.ru

Abstract
Singular Cauchy integral equations have been widely used for mathematical simulation of the actual physical and technical systems. They are considered universal at every level of simulation beginning with quantum field theory and up to strength analysis of the underground constructions. Therefore investigating system stability of such models under perturbation of their absolute terms and coefficients appears an urgent scientific task. The aim of the study is to show various aspects of stability of singular Cauchy integral sets of equations which are generalizing simulation models of the primal problems of the elasticity theory for homogeneous isotropic bodies. The methods of study are based on the properties of the Cauchy singular integral, on the general theory of Fredholm operators. When in use, systems of the singular integral equations are reduced to a set of Fredholm integral equations of the second kind and a set of the boundary value problems for analytic functions. The key results of the study are the following: development of the general determination method of the system index for singular integral equations, proof of the system stability against perturbations of the absolute terms of the set. Against perturbations of the boundary coefficients, the singular integral system is unstable. Demonstration of the stability of the singular integral Cauchy sets generalizing primal problems of the elasticity theory appears a significantly new result. The research of singular integral equations sets has been performed conducted on the space of functions satisfying the Holder condition. However the main research results prove to be true if we operate random functions converting in mean square. Stability of singular integral equations sets against perturbations of the absolute terms lays a foundation for calculus of approximations in real world tasks of defining the built-in stress of an elastic complex body.

Keywords:elasticity theory, singular integral equations, stability theory, boundary problems for analytic functions, singular Cauchy integral.

References

1. Balk M.B. (1991). Polyanalitic functions and their generalization (Polianaliticheskie funkcii i ih obobshcheniya). Results of science and technology VINITI (Itogi nauki i tekhniki VINITI). Ser. Sovr. Prob. matem. fund. napr. Vol.85. Moscow: VINITI. P. 187-246.
   2. Vekua I.N. (1970). On one solution of the primal biharmonic boundary value problem and Dirichlet problem (Ob odnom metode resheniya osnovnoj bigarmonicheskoj kraevoj zadachi i zadachi Dirihle). Some problems of mathematics and mechanics (Nekotorye probl. mat. i mekh.) Leningrad: Nauka. P. 120-127.
   3. Volodchenkov A.M., Yudenkov A.V. (2013). Primal problems of the elasticity theory with rectilinear anisotropy in the stochastic potential theory (Osnovnye zadachi teorii uprugosti tel s pryamolinejnoj anizotropiej v stohasticheskoj teorii potenciala). Proceedings: online journal of Kursk State University (Uchenye zapiski: elektronnyj nauchnyj zhurnal Kurskogo gosudarstvennogo universiteta). No. 2 (26). P. 14-17. URL: http://scientific-notes.ru/pdf/030-002.pdf.
   4. Volodchenkov A.M., Yudenkov A.V. (2007). Stability of the vector boundary value problem with shift simulating primal problems of the elasticity theory for an anisotropic body (Ustojchivost’ vektornoj kraevoj zadachi so sdvigom, modeliruyushchej osnovnye zadachi teorii uprugosti anizotropnogo tela). Survey of applied and industrial mathematics (Obozrenie prikladnoj i promyshlennoj matematiki). Issue 5-6. Moscow. P. 581-583.
   5. Gahov F.D. (1977). Boundary value problems (Kraevye zadachi). Moscow: Nauka. 640 p.
   6. Lekhnickij G.S. (1977). Elasticity theory of an anisotropic body (Teoriya uprugosti anizotropnogo tela). Moscow: Nauka. 416 p.
   7. Litvinchuk G.S. (1977). Boundary value problems and singular equations with shift (Kraevye zadachi i singulyarnye uravneniya so sdvigom). Moscow: Nauka. 448 p.
   8. Maksimova L.A., Yudenkov A.V. (2015). Theory of stochastic potential for the two-dimensional elasticity theory (Teoriya stohasticheskogo potenciala v ploskoj teorii uprugosti). Reporter of Yakovlev Chuvash State Pedagogical University (Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I.Ya. Yakovleva). Series: mechanics of the limiting state (Mekhanika predel’nogo sostoyaniya). No. 4 (26). P. 134-142.
   9. Maksimova L.A., Yudenkov A.V., Rimskaya L.P. (2016). Generalization of the singular Shermann integral equations with shift for the two-dimensional elasticity theory (Obobshchennye sistemy singulyarnyh integral’nyh uravnenij Shermana so sdvigom v ploskoj teorii uprugosti). Reporter of Yakovlev Chuvash State Pedagogical University (Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I.Ya. Yakovleva). Series: mechanics of the limiting state (Mekhanika predel’nogo sostoyaniya). No. 2 (28). P. 15-23.
   10. Muskhelishvili N.I. (1966). Certain primal problems of the mathematical elasticity theory (Nekotorye osnovnye zadachi matematicheskoj teorii uprugosti). Moscow: Nauka. 707 p.
   11. Muskhelishvili N.I. (1968). Singular integral equations (Singulyarnye integral’nye uravneniya). Moscow: Nauka. 511 p.
   12. Redkozubov S.A., Yudenkov A.V., Volodchenkov A.M. (2006). Simulating alteration of a linear elastic homogeneous body with the help of bianalytic functions (Modelirovanie processa linejnoj deformacii uprugogo odnorodnogo tela s pomoshch’yu bianaliticheskih funkcij). Reporter of Yakovlev Chuvash State Pedagogical University (Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I.Ya. Yakovleva), No. 1(48). Cheboksary. P. 128-137.
   13. Savin G.N. (1975). Stress distribution around holes(Raspredelenie napryazhenij okolo otverstij). Naukova dumka. Kiev.
   14. Sherman D.I. (1938). Statistic two-dimensional problem of the elasticity theory for isotropic inhomogeneous media (Staticheskaya ploskaya zadacha teorii uprugosti dlya izotropnyh neodnorodnyh sred). Proceedings of the seismological institute of the Academy of Sciences USSR (Tr. Sejsmol. in-ta AN SSSR). No. 86. P. 1-50.
   15. Sherman D.I. (1957). On one elasticity problem with mixed homogeneous conditions (Ob odnoj zadache teorii uprugosti so smeshannymi odnorodnymi usloviyami). Reports of the Academy of Sciences USSR (Dokl. AN SSSR), Vol. 114. No. 4. P. 733-736.
   16. Yudenkov A.V. (2002). Boundary value problems with shift for polianalytic functions and their application in the stochastic elasticity theory (Kraevye zadachi so sdvigom dlya polianaliticheskih funkcij i ih prilozheniya k voprosam staticheskoj teorii uprugosti). Smyadyn’: Smolensk. 268 p.
   17. Yudenkov A.V., Volodchenkov A.M. (2013). Primal problems of the elasticity theory for elastic two-dimensional anisotropic bodies in the stochastic potential theory (Osnovnye zadachi teorii uprugosti tel s pryamolinejnoj anizotropiej v stohasticheskoj teorii potenciala). Proceedings: online journal of Kursk State University (Uchenye zapiski. Elektronnyj nauchnyj zhurnal Kurskogo gosudarstvennogo universiteta). No. 2 (26). P. 14-17.
   18. Yudenkov A.V., Rimskaya L.P. (2016). Sets of singular integral equations in the theory of boundary value problems for bianalytic functions (Sistemy singulyarnyh integral’nyh uravnenij v teorii kraevyh zadach dlya bianaliticheskih funkcij). Current topics of the theory of partial equations (Aktual’nye problemy teorii uravnenij v chastnyh proizvodnyh). Abstracts of the international scientific conference in remembrance of A.V. Bicadze (Tezisy dokladov mezhdunarodnoj nauchnoj konferencii, posvyashchennoj pamyati akademika A.V. Bicadze). P. 146.
   19. Balk M.B. (1991). Polyanalytic functions. Berlin: Akademie Verlag. 192 p.

Information about authors:
Aleksey V. Yudenkov, Smolensk State Academy of Physical Culture, Sport and Tourism, Head of the Department of Management and Sciences, doctor of physics and mathematics, professor, Smolensk, Russia
Aleksandr M. Volodchenkov, Smolensk Branch of Plekhanov Russian University of Economics, Head of the Department of Sciences and Humanities, candidate of physics and mathematics, associate professor; Smolensk Branch of Saratov State Law Academy, assistant professor of the Department of Humanities, Socio-economical Sciences, Information Technologies and Law, Smolensk, Russia
Liliya P. Rimskaya, Smolensk Branch of Plekhanov Russian University of Economics, assistant professor of the Department of Management and Customs, candidate of physics and mathematics, associate professor, Smolensk, Russia