Approximate solution of the galerkin method for one non-classical problem of parabolic type
Abstract
The article considers one boundary value problem of parabolic type with a divergent main part, when the boundary condition contains the time derivative of the desired function. Such non-classical problems arise in a number of applied problems, for example, when a homogeneous isotropic body is placed in the inductor of an induction furnace and an electromagnetic wave falls on its surface. Such problems have been little studied, so the study of problems of parabolic type, when the boundary condition contains the time derivative of the desired function, is relevant. The work defines a generalized solution to the problem under consideration in the space The purpose of the study is to prove the theorem of the existence and uniqueness of an approximate solution of the Bubnov-Galerkin method for the considered non-classical parabolic problem with a divergent main part, when the boundary condition contains the time derivative of the desired function.
Keywords
Mixed problems, quasilinear equation, boundary conditionHow to Cite
References
Kaсur J. Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions// Math Slovoca, 1980, 30, N3, p. 213-237
Kaсur J. Nonlinear parabolic boundary valve problems with the time derivative in the boundary conditions// Lect Notes Math, 1979, 703, p. 170-178.
Mitropolskiy Yu. A., Nijnыx L. P., Kulchiskiy V.L. Nelineynie zadachi teploprovodnosti s proizvodnoy po vremeni v granichnom uslovii. –Preprint IM -74-15.Kiev.-1974.-32p.
Mixlin S.G. Chislennaya realizasiya variasionnyx metodov. M.-Nauka,-1966.-432 p.
Douglas J Jr, Dupont T. Galerkin methods for parabolic equations with nonlinear foundry conditions// NumerMalh.- 1973, 20, p. 213-237
Dench J. E., Jr, Galerking methods for some highly nonlinear problems// SIAM Numer anal, 1977, 14, p. 327-434.
Jutchell L. А Galerken method for nonlinear parabolic equations with nonlinear boundary conditions// SIAM J Numer anal 1979, 16, p. 254-299
Mamatov A.Z. Primeneniya metoda Galerkina k nekotoromu kvazilineynomu uravneniyu parabolicheskogo tipa// Vestnik LGU,-1981.-№13.-P.37-45.
Alisher Mamatov, Sayfiddin Bakhramov, Olim Abdurakhmonov, Dostonbek Abduraimov. Mathematical model for calculating the temperature of cotton in a direct-flow drying drum //AIP Conf. Proc. 2746, 060017 (2023)
Tikhinova I. M. (2016). Application of the stationary Galerkin method to the first boundary value problem for a mixed high-order equation// Mathematical notes of NEFU 2016, 23:4, 73-81.
Mamatov A.Z., Dosanov M.S., Raxmanov J., Turdibaev D.X. Odna zadacha parabolicheskogo tipa s divergentnoy glavnoy chastyu// НАУ (Natsionalnaya assotsiatsiya uchenyx). Ejemes. nauchnыy jurnal, 2020, №57, 1-chast, p.59-63.
Mamatov, A.Z., Narjigitov Х., Rakhmanov J., Turdibayev D. Refining the Galerkin method error estimation for parabolic type problem with a boundary condition E3S Web of Conferences 304, 03019 (2021) ICECAE 2021 https://doi.org/10.1051/e3sconf/202130403019
Mamatov, A., Parpiev, A., Shorakhmedova, M. Mathematical model for calculating the temperature field of a direct-flow drying drum. Journal of Physics: Conference Seriesthis link is disabled, 2021, 2131(5), 052067
Mamatov, A., Bakhramov, S., Narmamatov, A. An approximate solution by the Galerkin method of a quasilinear equation with a boundary condition containing the time derivative of the unknown function. AIP Conference Proceedingsthis link is disabled, 2021, 2365, 070003
Mamatov, A.Z., Usmankulov, A.K., Abbazov, I.Z., Norboyev, U.A., Mukhametshina, E.T. Determination of Temperature of Components of Cotton-Raw Material in a Drum Dryer with a Constant. IOP Conference Series: Earth and Environmental Sciencethis link is disabled, 2021, 939(1), 012052
Mamatov, A.Z., Pardaev, X.N., Mardonov, J.S.H., Plekhanov, A.F. Determining of the heat-moisture stateof raw cotton in a drum dryer. Izvestiya Vysshikh Uchebnykh Zavedenii, Seriya Teknologiya Tekstil'noi Promyshlennostithis link is disabled, 2021, 391(1), С. 46–49
Wheeler M.F. A priori error estimates for Galerkin approximation to parabolic partial differential equations. SIAM J. Numer. Anal.1973.-10.-P.723-759.
Ladyzhenskaya O.A., Solonnikov V.A., Uralseva N.N. Lineynye i kvazilineynye uravneniya parabolicheskogo tipa. M.-Nauka,-1967.-736 p.
Ladyzhenskaya O.A., Uralseva N.N. Lineynye i kvazilineynye uravneniya ellipticheskogo tipa. M.-Nauka,-1973.-576 p.
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