# Homchenko A. N., Valko N. V., Letvinenko O. I. The Kherson state university

 Название Homchenko A. N., Valko N. V., Letvinenko O. I. The Kherson state university Дата конвертации 21.06.2013 Размер 7.49 Kb. Тип Документы скачать >>>

MODELS OF A METHOD OF BARYCENTRIC AVERAGING

## The Kherson state technical university

The more difficultly the model, the is more than time and forces is spent for its creation and application. Therefore there is actual a question on simplification of algorithms of calculations, and consequently also models. The method of barycentric averaging uses properties of the center of weights of system of material points and also the theory of casual wanderings for averaging boundary potentials of a problem of restoration of function in internal points of area on its values on border of area.

There are two definitions of harmonious function - differential and integrated. Differential definition treats harmonious function as the decision of equation Laplace. Integrated definition of a harmonicity was offered to Kyobe in 1906. He has proved the theorem, that continuous in area G function u which accepts value which is equal to average arithmetic for any circle with the center in point Р which completely belongs G in each point Р of area, is harmonious in G.

In 1925 I.I. Privalov has been proved the theorem of harmonious function of three variables. Also it had been entered operators who allow to express a condition of a harmonicity of function without use of private derivatives.

Today integrated definition of harmonious function that uses not only curvilinear superficial integrals on border of a vicinity of point Р, and also double integrals on volume of a sphere. Besides the vicinity of point Р not necessarily should be circle or a sphere.

More often a presence of harmonious function connect з the decision of equation Laplace. But use of integrated definition of a harmonicity of function enables applications of the new approach to a task of restoration of function. The task is put to establish probability sense of integrated criterion of a harmonicity of function and on the basis of it{him} to consider construction of basic functions by a method of barycentric averagings.

Privalovs theorem enables to allocate area of a harmonicity function. If in this area to enclose other area function remain harmonious in new area and it will allow to consider discrete variants of integrated criterion of a harmonicity function.

On the basis of an integrated condition of a harmonicity function (I.I. Privalova's criterion) it is established, that property of integrated average has close communication{connection} with a principle of barycentric averagings of boundary potentials which is realized in single-step circuits of casual wanderings methods of Monte-Carlo.

The opportunity of averaging of models of bilinear interpolation on a square is investigated. Restriction of a degree interpolation a polynom allows to remove (or to reduce) not physical difference fields which arise at increase in amount of units at border.

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