THE SOLUTION PROBLEM OF DIRIHLE FOR EQUATION OF LAPLACE IN POLAR COORDINATES BY METHOD MONTE CARLO

Abstract

In this work, a model of random walks in polar coordinates for regions containing a coordinate pole is constructed. With this model, we can solve the problem Dirihle for equation of Laplace by method Monte-Carlo in a circle and in all areas that are part of the circle. Transient probabilities, as a rule, are found by the method of statistical tests, but in this work implemented the iteration procedure for calculating the a priori transition probabilities. We also note that the probabilities of transition to the boundary nodes, symmetric with respect to the axis from which the particle started, exactly coincide, which is practically not observed in the application of the statistical test method. Problem Dirihle for equation of Laplace for the on circle can be solved by using integral formula of Poisson. In this paper, a one-dimensional model of random walks in a circle is constructed on the basis of the Poisson integral formula. For a circle, a formula is derived for calculating the probability of transition from any node inside the circle to the node at the boundary of the circular region. Comparative testing of transition probabilities, calculated using two different models, was carried out. The results showed that for a circle, the one-time model of random walks always provides greater accuracy than multi-stage, provided that the distance between the nodes at the edge of the circle is twice less than that of the circle. The test results showed that the pole presense does not increase the error of the calculations. As a result, this multi-stage model can be used for any areas that contain a coordinate pole, can to build a iterative procedure for a more complex area, having the experience of constructing it for a circle and a semicircle. To calculate the transition probabilities, we must apply the formulas obtained. However, do not forget that the one-way model of random walks runs only in the circle.