Полигармоникалық оператор үшін Грин функциялары және тиянақты тарылулары
DOI:
https://doi.org/10.26577/JMMCS.2021.v109.i1.03Кілттік сөздер:
Уравнение Пуассона, полигармонические уравнения, задача Дирихле, задача Неймана, задача Робена, корректные сужения оператора.Аннотация
Бұл жұмыста классикалық – Дирихле, Нейман және Робен есептердің Грин функциялары айқын түрде көпөлшемді бірлік шарда Пуассон теңдеуі үшін көрсетілген. Пуассон теңдеуі үшін Дирихле есебінің Грин функциясын құрудың әртүрлі тәсілдері бар. Аудандардың көптеген түрлері үшін оның айқын түрі құрылған. Соңғы уақыттарда классикалық есептердің Грин функцияларын айқын түрде құруға деген қызығушылық қайта жандануды. Көпөлшемді шарда плигармоникалық теңдеу үшін Дирихле есебінің Грин функциясы айқын түрде құрылған, ал Нейман есебі үшін Грин функциясын құру ашық проблема болып қала беруде. Жұмыста көпөлшемді шарда полигармоникалық теңдеу үшін Дирихле есептерінің Грин функциясын құрудың тиімді әдісі көрсетілген. Дифференциалдық теңдеулер үшін жалпы тиянақты шекаралық есептерді табу әрқашан өзекті мәселе болып табылады. Бұл жұмыста операторлардың тарылуы мен кеңеюі теориясы қысқаша сипатталған және полигармоникалық оператор үшін тиянақты шеттік есептердің тарылулары көрсетілген.
Библиографиялық сілтемелер
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[3] Sobolev S.L., Vvedeniye v teoriyu kubaturnykh formul [Introduction to the theory of cubature formulas]. (Moscow: Nauka, 1974).
[4] Zorich V.A., Matematicheskiy analiz: Uchebnik. Chast’ II. [Mathematical Analysis: Textbook. Part II.] (Moscow:
Fizmatlit, 1984).
[5] Sadybekov M.A., Torebek B.T., Turmetov B.Kh., Representation of Green’s function of the Neumann problem for a multi-dimensional ball. Comp. Var. and Ell. Eq. – 2016. – 61:1. – 104–123.
[6] Sadybekov M.A., Turmetov B.Kh., Torebek B.T., On an explicit form of the Green function of the Roben problem for the Laplace operator in a circle. Adv. Pure Appl. Math. – 2015. – 6:3. 163–172.
[7] Kalmenov T.Sh., Koshanov B.D., Nemchenko M.Yu., Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere. Comp. Var. and Ell. Eq. – 2008. – 53:2. 177–183.
[8] Kalmenov T.Sh., Koshanov B.D., Representation for the Green’s function of the Dirichlet problem for the polyharmonic equations in a ball. Sib. Math. Journal. – 2008. – 49:3. 423–428. Doi: 10.1007/s11202-008-0042-8
[9] Kalmenov T.Sh., Suragan D., On a new method for constructing the Green’s function of the Dirichlet problem for a polyharmonic equation. Differen. eq. – 2012. – 48:3. – 435–438. Doi: 10.1134/S0012266112030160
[10] Begehr H., Biharmonic Green functions. Le matematiche. – 2006. – 61. – 395–405.
[11] Wang Y., Ye L., Biharmonic Green function and biharmonic Neumann function in a sector. Comp. Var. and Ell. Eq. – 2013. – 58:1. – 7–22.
[12] Wang Y. Tri-harmonic boundary value problems in a sector. Comp. Var. and Ell. Eq. – 2014. – 59:5. – 732–749.
[13] Begehr H., Du J., Wang Y., A Dirichlet problem for polyharmonic functions. Ann. Mat. Pura Appl. – 2008. – 187:4. – 435–457.
[14] Begehr H., Vaitekhovich T., Harmonic boundary value problems in half disc and half ring. Funct. Approx. Comment. Math. – 2009. – 40:2. – 251–282.
[15] Begehr H., Vaitekhovich T., Modified harmonic Roben function. Comp. Var. and Ell. Eq. – 2013. – 58:4. – 483–496.
[16] Neumann J.von., Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. – 1929. – 102. – 49–131.
[17] Vishik M.I., Ob obshchikh krayevykh zadachakh dlya ellipticheskikh differentsial’nykh uravneniy [General boundary value problems for elliptic differential equations]. Trudy Mat. O-va. – 1952. – 3. – 187–246.
[18] Vishik M.I., Boundary value problems for elliptic equations degenerating on the boundary of the domain. Matem. Sbor. – 1954. – 7:3. – 1307–1311.
[19] Bitsadze A.V., Samarskyi A.A., On some simplest generalizations of linear elliptic boundary value problems. Dokl. Akad. Nauk SSSR. 1969. – 185:4. – 739–740.
[20] Dezin A.A. Partial differential equations. Berlin etc. Springer-Verlag, 1987.
[21] Kokebaev B.K., Otelbaev M., Shynybekov A.N., On the theory of restriction and extension of operators. I. Izves. AN KazSSR. Ser. fiz-mat. 1982. – 5. – 24–27.
[22] Kokebaev B.K., Otelbaev M., Shynybekov A.N., On the theory of restriction and extension of operators. II. Izves. AN KazSSR. Ser. fiz-mat. – 1983. – 1. – 24–27.
[23] Kokebaev B.K., Otelbaev M., Shynybekov A.N., On the expansion and restriction of operators. Dokl. Akad. Nauk SSSR. – 1983. – 271:6. – 1307–1311.
[24] Oynarov R., Parasidi I.N., Correctly Solvable Extensions of Operators with Finite Defects in a Banach Space. Izves. AN KazSSR. Ser. fiz-mat. – 1988. – 5. – 35–44.
[25] Burenkov V.I., Otelbaev M., On the singular numbers of correct restrictions of non-selfadjoint elliptic differential operators. Eurasian Math. J. 2011. – 2:1. – 145–148.
[26] Koshanov B.D., Otelbaev M., Correct Contractions stationary Navier-Stokes equations and boundary conditions for the setting pressure. AIP Conf. Proc. 2016. – 1759. /10.1063/1.4959619
[27] Kanguzhin B.E., Changes in a finite part of the spectrum of the Laplace operator unter delta-like perturbations. Differen. Eq. 2019. – 55:10. – 1428–1335.
[28] Kanguzhin B.E., Tulenov K.S., Singular perturbations Changes of Laplace operator and their recolvents. Comp. Var. and Ell. Eq. 2020. – 65:9. – 1433–1444.
[29] Biyarov B.N., Svistunov D.L., Abdrasheva G.K., Correct singular perturbations of the Laplace operator. Eurasian Mathematical Journal. 2020. – 11:4. – 25–34.
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Как цитировать
Koshanov, B. D. (2021). Полигармоникалық оператор үшін Грин функциялары және тиянақты тарылулары. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 109(1), 35–55. https://doi.org/10.26577/JMMCS.2021.v109.i1.03
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