THE PROBLEM OF POTENTIAL THEORY FOR A LAYER WITH A CIRCULAR CYLINDRICAL CAVITY

The generalized Fourier method for the solution of the boundary value problems of the mathematical physics is applied to the solution of the Dirichlet problem for the Laplace equation in a plane-parallel layer with a cavity in the form of a circular cylinder, whose generatrix is parallel to the boundary planes of the layer. The formulas for the expansion of harmonic functions from the Cartesian coordinate system to the cylindrical coordinate system and their inverses, which were used in the work, allowed to reduce the problem to the infinite system of linear algebraic equations, for which the possibility of the finding an approximate solution by the reduction method is proved under the condition of non-touching surfaces. In this case, the approximate solutions tend to the exact solution with increasing the order of the truncated systems. The method can be extended to other basic problems of potential theory for one or several cylindrical inclusions, and to the basic problems of the spatial theory of elasticity.