Block elements with a cylindrical boundary in macro- and nanostructures

Babeshko, V. A.; Евдокимова, О. В.; Babeshko, O. M.

doi:10.1134/s1028335811100119

Cited by 7 publications

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“…This theorem is related to the represen This theorem explains the possibility of choosing a rich arsenal of every possible region admitted by the accepted topological structure and a particular bound ary value problem as the carriers of block elements. Each block element carrier can have its own local sys tem of coordinates, the relation of which with the local systems of carriers of neighboring blocks is controlled by a map [12,13].…”

Section: Theorem 1 the Set Of Block Elements Of The Bound Ary Value mentioning

confidence: 99%

Certain general properties of block elements

2012

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The results of determination of block elements proposed in [1][2][3][4] mainly for boundary value prob lems with constant coefficients give the methods of their construction, from which it is possible to see the close relation of these elements to a particular bound ary value problem. This circumstance induces the problem on a possible restriction of using block ele ments that have their origin in particular boundary value problems.In this paper, we present results describing the rela tion between the block elements for different bound ary value problems, which show that important rela tions can exist between the block elements of these boundary value problems. The listed set of properties of block elements enables us to use them more widely in various fields.1. The possibilities of the block element method are displayed by its use in a number of polytypic prob lems presented below.In [1][2][3][4], the concept of a block element is intro duced, and a number of examples of particular block elements are given for certain boundary value prob lems. It is shown that the block elements are deter mined by the boundary value problem and can always be constructed for an unambiguously solvable bound ary value problem formulated for a set of partial dif ferential equations of a finite order with constant coef ficients in the region with a piece smooth boundary [5][6][7]. They also can be constructed for the boundary value problems with variable coefficients admitting the separation of variables [8].In the general case of the boundary value problems with variable coefficients, their region of formulation of the boundary value problem is divided by a mesh for using the block element method. The mesh should be so dense that it could be possible to consider the coefficients in a division cell as constant [1][2][3][4]9].A certain practice of applying the block elements shows that their use simplifies the formulation of a number of boundary value problems and also the con struction of their solutions. For example, the block element method makes it possible to solve the bound ary value problems for homogeneous and inhomoge neous sets of partial differential equations in a similar way [7]. For its use, it is unnecessary to construct indi vidually the general solutions of homogeneous differ ential equations and the partial solutions of inhomo geneous equations with the subsequent fulfillment of boundary conditions. In the unsteady boundary value problems, the block element method raises both the edge boundary conditions for sets of partial differential equations and the initial conditions of a boundary value problem [10] to the rank of boundary condi tions; i.e., the initial conditions in the block element method become the boundary conditions. The block element method makes it possible to consider the same boundary value problems in the bounded, semi bounded, and unbounded regions.The block elements enable us to simplify the deri vation of certain important characteristics of the solu tion.For example, the block elements describe t...

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Section: Theorem 1 the Set Of Block Elements Of The Bound Ary Value mentioning

confidence: 99%

Certain general properties of block elements

2012

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“…Precisely, these are the Dirichlet, Neumann, or mixed boundary conditions that are considered the same as in a previous study [1].…”

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“…The block ele ments have a number of advantages, whose possibili ties can considerably contribute to the investigation of solutions of the differential equations [3]. It is neces sary to note that the block element method and the differential and integral factorization methods [1,4,5] do not repeat the approach based on the automor phism and related to the Lie group analysis [6][7][8]. The purpose of the first ones is the construction of the solu tions of boundary value problems for large sets of par tial differential equations, while the last one is aimed at construction of solutions for linear and nonlinear differential equations disregarding the boundary con ditions.…”

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“…Using the differential factorization method, we construct the block elements for the inner and outer boundary value problem for a cylinder with a compli cated boundary, a simpler case of which is considered in [1]. The methods of the manifold automorphism implementation described in [2] are at the basis of the construction.…”

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“…Applying transforms (4) to Eq. (3), we construct the outer form [1], which has the following form in the coordinates :…”

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