Mamuli Zakradze

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 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

The Dirichlet ordinary and generalized harmonic problems for some 3D finite domains are considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. An algorithm of numerical solution by the method of probabilistic solution (MPS) is given, which in its turn is based on a computer simulation of the Wiener process. Since, in the case of 3D generalized problems there are none exact test problems, therefore, for such problems, the way of testing of our method is suggested. For examining and to illustrate the effectiveness and simplicity of the proposed method five numerical examples are considered on finding the electric field. In the role of domains are taken ellipsoidal, spherical and cylindrical domains and both the potential and strength of the field are calculated. Numerical results are presented.

A Dirichlet generalized harmonic problem for finite right circular cylindrical domains is considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. It is shown that if a finite domain is bounded by several surfaces and the curves are placed in arbitrary form, then the generalized problem has a unique solution depending continuously on the data. The problem is considered for the simple case when the curves of discontinuity are circles with centers situated on the axis of the cylinder. An algorithm of numerical solution by a probabilistic method is given, which in its turn is based on a computer simulation of the Wiener process. A numerical example is considered to illustrate the effectiveness and simplicity of the proposed method.

An algorithm for the approximate solution of the Dirichlet generalized problem is proposed. The term “generalized” indicates that the boundary function has a finite number of first kind break points. The solution consists of the following stages:

1) the reduction of the Dirichlet generalized problem to an ordinary auxiliary problem for a harmonic function; 2) an approximate solution of the auxiliary problem by the modified version of the MFS (the method of fundamental solutions); 3) the construction of an approximate solution of the generalized problem from the solution of the auxiliary problem. An example is considered in which the break points are the cusp ones.

An algorithm for an approximate solution of definite type Dirichlet generalized problem is given. It consists of the following stages: 1) reduction of the Dirichlet generalized problem to an ordinary new (auxiliary) problem for harmonic function; 2) approximate solution of the new problem by the modified version of MFS (the method of fundamental solutions); 3) definition of the approximate solution of the posed generalized problem by the solution of the new problem. Examples of application of the proposed algorithm and the results of numerical experiments are given.

This paper focuses on the actual practical information regarding boundary problems with the boundary singularities. Dirichlet generalized boundary problem for a Laplace equation is considered for both finite and infinite domains. The case of a boundary function with a limited number of first kind break points is considered under this generalized boundary problem. In this paper, one method is a given ordinary problem for the reduction of the Dirichlet generalized boundary problem with an harmonic function. The method is constructed on the basis of fictitious sources and is applied to finite and infinite domains. From our point of view, this method is characterized by simplicity, stability, and allows for high accuracy. Also, it is oriented for a wide range of users (especially for the researchers in engineering problems). Examples with cusp points in the role of break points are also considered. The results of numerical experiments are given.

In this paper we investigate the question how the method of conformal mapping (MCM) can be applied for approximate solving of the generalized Dirichlet boundary problem for harmonic function. Under the generalized problem is meant the case when a boundary function has a finite number of first kind break points. The problem is considered for finite and infinite simply connected domains. It is shown that the method of fundamental solutions (MFS) is ineffective for solving of the considered problem from the point of view of the accuracy. We propose an efficient algorithm for approximate solving of the generalized problem, which is based on the MCM. Examples of application of the proposed algorithm and the results of numerical experiments are given.

It is pointed out that when approximate conformal mappings are applied to practical problems the univalence of their mapping functions in general should be investigated. Simple examples show that a high accuracy of the approximate mapping functions does not necessarily mean the univalence of them, which may cause a certain difficulty in the problem solving.

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.

An algorithm for the determination of elastico-dynamic state of homogeneous isotropic elastic body with a free surface over finite interval of time is given. In the present paper we consider the case,when the mentioned state of the elastic body is caused by the action of a simple concentrated force applied to a fixed point of the body, with the force varying in time by a non-periodical law. The realization of the algorithm is based on the method of fundamental solutions. The effectiveness of the proposed algorithm in comparison with the methods of integral transformation and Green’s function is shown. For illustration of efficiency, an example has been discussed

A matrix of fundamental solutions of the equation of dynamic elasticity theory is

constructed for a homogeneous isotropic medium in the case, when a concentrated

force changes at time exponentialy-periodicaly. Properties of the indicated matrix and its corresponding stress tensor are investigated. Besides, a way of calculation of their elements is given.

The Dirichlet generalized problem for the Laplace equation in the case of a finite

m-connected domain D which lies in the plane z = x + iy is considered. Under the Dirichlet generalized problem is meant the problem when a boundary function has a finite number of first kind points of discontinuity. It is shown that the method of fundamental solutions (MFS) is not suitable for solving of the considered problem. To avoid this situation it is recommended to smooth preliminary the boundary function, i.e., to reduce the generalized problem to an ordinary problem and to solve the latter by the M F S method. Analytic forms of smoothing functions for a finite simply and multiply connected domains are given.

Numerical examples are considered to illustrate effectiveness and simplicity of the proposed way.

In the present paper, the question of solution of the Dirichlet boundary problem for the Poisson equation by the method of conformal mapping (MCM) is considered. It is shown that application of the method is especially effective in the case where a particular solution to the Poisson equation cannot be written explicitly. The cases of both finite and infinite domains are considered. Illustrative examples are given.

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.

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.

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.

A modified system of fundamental solutions of the Laplace operator is constructed

in the plane case. Based on this system, the method of an approximate solution of

Dirichlet plane external boundary value problem is developed. Numerical examples are considered to illustrate high accuracy and simplicity of the method.

In this paper, for determination of the electric and thermal stationary fields the Dirichlet ordinary and generalized harmonic problems are considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. For numerical solution of boundary problems the method of probabilistic solution (MPS) is applied, which in its turn is based on a modeling of the Wiener process. The suggested algorithm does not require an approximation of a boundary function, which is main of its important properties. For examining and to illustrate the effectiveness and simplicity of the proposed method four numerical examples are considered on finding the electric and thermal fields. In the role of domains are taken: finite right circular cone and truncated cone; a rectangular parallelepiped. Numerical results are presented.

The algorithm on solving the Internal three-dimensional Dirichlet boundary value problem for a harmonic function by the method of probabilistic solution is given. The algorithm is based on a computer simulation of the Wiener process. To illustrate the effectiveness and simplicity of the proposed method a numerical examples considered.