Stability of the Galerkin Method for one Quasilinear Parabolic Type Problem

  • Mamatov Alisher Zulunovich The Tashkent Institute of Textile and Light Industry, (100100, Tashkent, Shokhjakhon street, 5), Uzbekistan
  • Narjigitov Xusanboy Departmentof Mathematics, Gulistan state University of Uzbekistan, Uzbekistan
  • Jamuratov Kengash Departmentof Mathematics, Gulistan state University of Uzbekistan, Uzbekistan
  • Egamqulov Abbos Ismatulla oʻgʻLi Departmentof Mathematics, Gulistan state University of Uzbekistan, Uzbekistan
  • Rakhmanov Jamshidbek Turdaliyevich Departmentof Mathematics, Gulistan state University of Uzbekistan, Uzbekistan
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

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.

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Published
2021-06-22
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
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Articles