ზაზა თაბაგარი

The method of reduction of the Dirichlet generalized boundary value problem for a harmonic function to an ordinary problem is given in the case of finite multiply connected and infinite domains. The method is constructed on the basis of the scheme suggested by M. A. Lavrent’ev and B. V. Shabat, which can be applied only to a finite simply connected domain. Examples are considered and the results of numerical calculations are given.

The algorithm of solving the external three-dimensional Dirichlet boundary value problem for a harmonic function by the probabilistic method is given. The algorithm consists of the following stages: 1) transition from an infinite domain to a finite domain by an inversion; 2) consideration of a new boundary problem on the basis of Kelvin’s theorem for the obtained finite domain; 3) application of the probabilistic method to solving a new problem, which in turn is based on a computer simulation of the Wiener process; 4) definition of the solution of the statement problem for the infinite domain by the solution of the new problem. For illustration an example is considered.

The present paper is devoted to the analysis of an explicit analytic solution of the Dirichlet generalized harmonic problem for a finite right circular axisymmetric cylindrical ring. We intend to use it for testing. For construction of the mentioned solution, the following methods are applied: separation of variables, particular solutions and heuristic method. Since the heuristic method does not guarantee finding the best solution, because of this, properties of the noted solution were investigated. It is shown that the above-mentioned problem can be used in the role of a test with the help of the given analytic solution

In this paper, we describe how the probabilistic method (PM) can be applied to numerical solving the Dirichlet generalized harmonic problem in regular n-sided pyramidal domains. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. In the considered case, the edges of the pyramid represent the curves. Application of the PM consists of the following main stages: a) computer modelling of the Wiener process; b) construction of an algorithm for finding the point of intersection of the trajectory of the simulated wiener process and the surface of the pyramid; c) checking a scheme and a corresponding calculating program needed for numerical implementation and reliability of the obtained results; d) finding a probabilistic solution of generalized problems at any fixed points of the considered pyramid. The algorithm does not require approximation of a boundary function, which is main of its important properties. For illustration of the effectiveness and simplicity of the suggested method several numerical examples are considered and numerical results are presented.

The algorithm of a probabilistic solution to the plane Dirichlet problem is given for the Laplace equation in the case of an infinite plane with a hole. A method of a computer simulation of diffusion process based on "white noise" is offered. It is used conformal mapping. For illustration an example is considered.

Two ways of probabilistic solution of the Dirichlet generalized boundary problem for harmonic functions are considered: a) Direct application of the probabilistic method; b) Reduction of generalized problem to the ordinary problem and solution of the latter by the probabilistic method. Investigation showed the reliability of results of both methods, which in turn indicates the superiority of direct application. The simplicity and the possibility of direct application of this method also indicate the superiority against the other well-known methods.