NUMERICAL SOLUTION OF THE WEBER PROBLEM WITH THE USE OF SPLINE APPROXIMATION

Abstract

In computer math systems for the description of different types of interpolation splines the fragmentary method is used. In the tasks of mathematical physics, when searching the solution in the spline form, the traditional way for the presenting of latter leads to a significant slowdown in the implementation of the corresponding numerical methods. From the spline theory, it is known that an interpolation spline can be described by one formula – a linear combination of basic functions. This article is devoted to the comparative analysis of the impacts of these forms of the interpolation splines representation on the characteristics of the solution of the Weber problem in a polar coordinate system by the Bubnov-Galyorkin method. Weber's historic problem about the torsion of a cylindrical shaft with a circular twist currently is still used as a test problem for the approbation of new numerical methods for solving second-order elliptic problems owing to the existence of an exact solution. The proposed solution is presented as a product of a two-dimensional spline and an auxiliary factor. The auxiliary factor is the implicit equation of the domain boundary. In this way the fulfillment of the requirements for the basic functions in the Bubnov-Galyorkin method, namely, the satisfaction of the boundary conditions in the investigated problem, is ensured. Onedimensional splines (each on its polar coordinate) are B-splines. To describe the basic functions of these B-splines, two exploratory forms of representation are used. The solution of this task is carried out by means of the system of computer mathematics Maple. The timing of the implementation of the algorithm for solving the Weber problem by the Bubnov-Galyorkin method on a polar grid is carried out. The practical convergence of the Bubnov-Galyorkin method was also evaluated for the approximate performance of the necessary operations of integration and its comparison with the theoretical rate of convergence of the method is completed. As a result of the performed research it is practically shown that in the systems of symbolic mathematics the use of the description splines as the only expression has certain advantages.