Restoration of the polygon vertices using middle of it’s sides

Abstract

Many important technical problems lead to the solution of boundary value problems of differential equations with partial derivatives. With the help of boundary value problems, it is possible to describe such processes as heat and mass transfer, diffusion, fluid flow, propagation of acoustic waves, electromagnetism, deformation of a solid. Some boundary-related problems can be solved analytically. Usually in these cases the geometry of the region and boundary conditions are relatively simple, and the equations with partial derivatives are linear. In practical problems arising in engineering and applied sciences, it is difficult to rely on obtaining analytical solutions, even if the differential equations are linear, since these problems are characterized by extreme irregularity of the boundaries of the regions and (or) heterogeneity of the material; since the solution of problems can not be constructed with the help of simple mathematical functions. In such cases, looking for approximate numerical solutions. Let the coordinates of the points, which are the middle of the sides of the N-corner, are known. Need to find the coordinates of the vertices of the polygon. Such problems arise in the numerical solution of singular and hypersingular integral equations using the method of collocation. Note that the problem has a unique solution for an odd number of sides. With a pair of sides, the matrix of the system is degenerate, which leads to the need to impose additional conditions for the solution. In the paper we consider a problem of flat polygon construction using the coordinates of its side’s centers. These problems arise in numerical solution of singular and hypersingular integral equations using the discrete singularities methods. The special shift matrixes were introduces to get the analytical solution of coordinates of polygon vertices. The problem has the only decision if the number of sides is odd. There was given the example of recovering the square vertices by the seven node's values. These results can help in construction of the calculation area if we haven’t full information about it.