M. R. Timerbaev

Kazan (Volga Region) Federal University, Kasan, Tatarstan, Russia

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Publications (10)1.66 Total impact

  • M. R. Timerbaev, N. V. Timerbaeva
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    ABSTRACT: Sobolev spaces with weights taking infinite values at some interior points of a two-dimensional domain are considered. For functions from these spaces, a Hardy inequality is obtained. Embedding theorems for weighted Lebesgue spaces and equivalent renorming theorems are proved.
    Lobachevskii Journal of Mathematics 10/2013; 34(4). DOI:10.1134/S1995080213040069
  • A. A. Sobolev, M. R. Timerbaev
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    ABSTRACT: In this paper we construct a high accuracy variant of the finite element method for an ordinary differential equation of the fourth order whose coefficients are degenerate on the boundary. The proposed technique is based on the multiplicative and additive-multiplicative separation of singularity. We prove that the convergence rate of the proposed technique is optimal in a given class of smoothness of right-hand sides. Keywords and phrasestwo-point boundary value problem–finite element schemes–weight function spaces–multiplicative and additive-multiplicative separation of singularity
    Russian Mathematics 05/2011; 55(5):74-77. DOI:10.3103/S1066369X11050136
  • A.A. Sobolev, M.R. Timerbaev
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    ABSTRACT: Two-point boundary value problem for a differential equation of fourth-order with degeneration is considered. This problem is solved by the finite element method of high-order accuracy with a multiplicative separation of singularity. The optimal convergence rate of the presented method for a given class of smoothness of the right-hand sides is proved.
    01/2010; 1(1).
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    A. D. Lyashko, Sh. I. Tayupov, M. R. Timerbaev
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    ABSTRACT: In this paper for the finite elementmethod for systems of degenerate elliptic equations we develop high-accuracy schemes based on multiplicative singularity extraction. We prove theorems about the smoothness of a solution. Based on these theorems we estimate the error of the proposed method.
    Russian Mathematics 07/2009; 53(7):17-27. DOI:10.3103/S1066369X09070032
  • A. D. Lyashko, M. R. Timerbaev
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    ABSTRACT: The paper deals with the numerical solution of a generalized spectral boundary value problem for an elliptic operator with degenerating coefficients. We suggest an approximate method based on the multiplicative separation of the singularity, whereby the eigenfunctions are approximated by piecewise linear functions multiplied by a weight specially chosen depending on the order of degeneration of the coefficients. For this method, we obtain error estimates justifying its optimality.
    Differential Equations 07/2008; 44(7):999-1005. DOI:10.1134/S0012266108070124 · 0.42 Impact Factor
  • S.I. Tayupov, M.R. Timerbaev
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    ABSTRACT: The purpose of the paper is construction of a finite element schemes with a high approximation order for two-pointed heterogeneous boundary-value problem with degenerated coefficients, based on a multiplicative allocation of singularities. It is proved, that this method has optimal order of convergence rate.
    01/2006; 4(4).
  • M. M. Karchevskii, AD Lyashko, M. R. Timerbaev
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    ABSTRACT: In the present paper, we suggest and investigate a mixed finite-element scheme for quasilinear fourth-order elliptic equations degenerating on the domain boundary. For example, equations appearing in the description of elastoplastic deflections of thin plates of variable thickness (e.g., see [1, p. 195; 2, p. 69]) are special cases of such equations. The structure of our schemes is close to that studied in [3] (see also [4]). We use products of second-order derivatives of the desired function by a weight function determining the character of the equation degeneration as auxiliary unknowns in the mixed statement of the problem. This permits us to construct and investigate schemes of an arbitrary-order accuracy on the basis of Lagrangian elements in a unified way. A number of conformal schemes for quasilinear degenerating fourth-order elliptic equations based on classical affine-equivalent or almost affine-equivalent sets of finite elements of the class C-(1) [5, pp. 325-353 of the Russian translation] were investigated in [6].
    Differential Equations 06/2000; 36(7):1050-1057. DOI:10.1007/BF02754507 · 0.42 Impact Factor
  • M. R. Timerbaev
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    ABSTRACT: We suggest a numerical method for degenerate equations in which the solution u(x) is represented in the boundary layer region in the form of the product u(x) = a(x)~o(x), where the function a(x) is constructed with the use of the coefficients of the degenerate differential operator and takes account of the asymptotics of the solution in a neighborhood of the degeneration points and the function ~(x) is much more suitable for FE approximations than u(x). The approximate solution is sought in the form ~th(X ) ~- O'(X)~gh(X), where ~h(X) is an FE approximation to p(x) on a quasiuniform grid. The performance characteristics of the method (such as accuracy, programmability, and time and memory requirements) are the same as in the regular case.
    Differential Equations 06/2000; 36(7):1086-1093. DOI:10.1007/BF02754511 · 0.42 Impact Factor
  • A.D. Lyashko, M.R. Timerbaev
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    ABSTRACT: From the introduction: The article is devoted to investigation of existence and uniqueness of generalized solution of a linear degenerate elliptic equation of order 2m for two cases. First, for the case where the equation degenerates on a part of the domain’s boundary (in particular, on the whole boundary). Second, when the degeneration occurs inside the domain. We analyze the problem’s statement in dependence on equation coefficients’ degeneration degree by application of the embedding theorem of the weight Sobolev spaces. For approximate solution we suggest the finite element method.
    Russian Mathematics 01/1999; 43(5).
  • M.M. Karchevskij, A.D. Lyashko, M.R. Timerbaev
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    ABSTRACT: The authors propose a finite element method in order to solve some fourth-order quasilinear elliptic boundary value problems which degenerate on the boundary of the domain. They follow the classical lines of analysis in the finite element method (existence and uniqueness of a generalized solution, interpolation results in weighted Sobolev spaces, etc.). Consequently, they obtain estimations relating the accuracy of the method with the smoothness of the solution, the degeneration degree and the roughness of the mesh nearby the degeneration line.
    Differential Equations 35(2). · 0.42 Impact Factor

Publication Stats

12 Citations
1.66 Total Impact Points

Institutions

  • 2011
    • Kazan (Volga Region) Federal University
      Kasan, Tatarstan, Russia
  • 2000
    • Kazan State Medical University
      Kasan, Tatarstan, Russia