Article Sidebar

Published
Jun 22, 2021
Download
PDF
Statistic
Read Counter : 77 Download : 60

Main Article Content

Abstract

The article considers a parabolic-type boundary value problem with a divergent principal part, when the boundary condition contains the time derivative of the required function. An approximate solution is constructed and the stability of the Galerkin method of the problem under consideration is established. In this paper, a generalized solution of the problem under consideration is defined in the spaceThe proposed boundary value problem is considered under certain conditions for the function involved in the equation and the boundary condition, which allow the existence and uniqueness of the generalized solution. For the numerical solution of the problem under consideration, an approximate solution was constructed using the Bubnov-Galerkin method. The concept of stability of the Galerkin process for this problem is introduced. The aim of the research is to obtain a condition for the stability of the computational process of the considered mixed problem.Using the Bubnov-Galerkin method, the problem under consideration is reduced to solving a system of ordinary differential equations. Further, we consider the "perturbed" problem for the system of the Bubnov-Galerkin method and obtain estimates for the difference between the solutions of the original and perturbed systems. The article establishes the stability of the Galerkin method of the problem under consideration, under the conditions of strong minimality of the coordinate system, which allows the calculation of an approximate solution of the problem under consideration by the proposed Bubnov-Galerkin method.

Keywords

Mixed problems quasilinear equation boundary condition Galerkin method generalized solution parabolic type approximate solution error estimate a priori estimates coordinate system monotonicity stability strongly minimality inequalities boundary domain dot product continuity error

Article Details

How to Cite
Zulunovich, M. A., Xusanboy, N., Kengash, J., Ismatulla oʻgʻLiE. A., & Turdaliyevich, R. J. (2021). Stability of the Galerkin Method for one Quasilinear Parabolic Type Problem. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 2(6), 6-12. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/90

References

  1. [1] Kaсur J. Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions// Math Slovoca, 1980, 30, N3, p 213-237
  2. [2] Kaсur J. Nonlinear parabolic boundary valve problems with the time derivative in the boundary conditions// Lect Notes Math, 1979, 703, p. 170-178.
  3. [3] MitropolskiyYu.A., Nijnykh L.P., Kulchitskiy V.L. Nelineyniyezadachiteploprovodnosti s proizvodnoypovremani v granichnomuslovii. –Preprint IM -74-15.Kiev.-1974.p.32.
  4. [4] Mixlin S.G. Chislennayarealizatsiyavariatsionnykhmetodov. M.-Nauka,-1966.-p.432.
  5. [5] Douglas J Jr, Dupont T. Galerkin methods for parabolic equations with nonlinear foundry conditions// NumerMalh.- 1973, 20, p. 213-237
  6. [6] Dench J. E., Jr, Galerking methods for some highly nonlinear problems// SIAM Numer anal, 1977, 14, p. 327-434.
  7. [7] Jutchell L. АGalerken method for nonlinear parabolic equations with nonlinear boundary conchtions// SIAM J Numer anal 1979, 16, p. 254-299
  8. [8] Tikhinova I.M., “Application of the stationary galerkin method to the first boundary value problem for a mixed high-order equation”, Mathematical notes of NEFU 23:4, p.73-81(2016).
  9. [9] Fedorov V.E., “The stationary galerkin method applied to the first boundary value problem for a higher order equation with changing time direction”, mathematical notes of NEFU 24:4, 67-75 (2017).
  10. [10] Mamatov A.Z. PrimeneniyametodaGalerkina k nekotoromukvazilineynomuuravneniyuparabolicheskogotipa\\Vestnik LGU,-1981.-№13.-p.37-45.
  11. [11] Mamatov A.Z., Djumabaev G. Ob odnoyzadacheparabolicheskogotipa s divergentnoyglavnoychastyu // 53 mejdr. Nauchnopraktiches. konf., VGTU, Vitebsk, R.Belaruss» 2020 y.
  12. [12] Mamatov A.Z., Baxramov S. Priblizhennoeresheniemetodagalerkinakvazilineynogouravneniya s granichno’musloviem, soderzhahiyproizvodnuyupovremeniotiskomoyfunktsii // Uzbekistan -Malaysiya//A collection of scientific articles International Scientific Online Conference, NUz, 24-25 august 2020 y.,p.239
  13. [13] Mamatov A.Z., Dosanov M.S., Raxmanov J., Turdibaev D.X. Odnazadachaparabolicheskogotipa s divergentnoyglavnoychastyu // NAU (Natsionalnayaassotsiatsiyauchenyx). Ejemes. nauchniyjurnal, 2020, №57, 1-chast, p.59-63.
  14. [14] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N. Lineynyeikvazilineyniyeuravneniyaparabolicheskogotipa. Moscow.-Nauka,-1967.- p.736
  15. [15] Ladyzhenskaya O.A., Uraltseva N.N. Lineynyeikvazilineyno’euravneniyaellipticheskogotipa. Moscow.-Nauka,-1973.- p.576
  16. [16] Mamatov A.Z., Axmatov N. Chislennoyeresheniezadachiopredeleniyateplo-vlajnogosostoyaniyakhlopka-syrtsa v barabannoy sushilke1 //Journal. Textile problems.-2016.-№3,-p.80-86
  17. [17] Mamatov А., Zulunov R. Sodikova М. Application Of Variational Grid Method For The Solution Of The Problem On Determining Mosture Content Of Raw Cotton In A Drum Dryer// The American Journal of Engineering and Technology (ISSN – 2689-0984) Published: February 26, 2021 | Pages: 75-82 Doi: https://doi.org/10.37547/tajet/Volume03Issue02-11
  18. [18] Polyanin A. D., Schiesser, W. E. and Zhurov A. I. Partial differential equations (2008), Scholarpedia, 3(10):4605.
  19. [19] Mamatov A.Z., Djumabaev G. Ob ustoychivostipriblijennogoresheniyaodnoyzadacheopredeleniyateplo-vlajnogosostoyaniyaxlopka-syrtsa // Journal. Problemytekstilya. Uzbekistan,2010.-№2, p. 86-90
  20. [20] Mamatov A.Z., Atajanova M. Ob ustoychivostimetodaGalerkinadlyaodnoyzadachiteploperenosa // A collection of scientific articles, AndijanSU, Uzbekistan, 2016. p.121-124
  21. [21] Burkhoff G., Schultz M. H., Varda R.S. Piecewise Hermite interpolations in one and two variables with applications to partial differential equations// Numer Math, 1968, 11,p. 232-256.
  22. [22] Bramble J.H., Hilbert S. R. Bounds for a class of linear functionals with applications to Hermite interpolation //Numer Math, 1971, 16, 362-369.
  23. [23] Pimenov V. G., Lozhnikov A. B. CHISLENNIYE METODY. Ekaterinburg IzdatelstvoUralskogouniversiteta,Russia/ p.2014.-106
  24. [24] Chislennyemetody: uchebno-metodicheskoyeposobiye / P.A. Denisov, V.F. Petrov; Yuzhno-Rossiyskiygosudarstvenniypolitexnicheskiyuniversitet (NPI) imeni M.I. Platova. Novocherkassk: YURGPU(NPI).– 2017. –p. 64