Heat conduction of 2D composite materials with symmetric inclusions: A model and reduction to a vector-matrix problem

We consider steady potential heat conduction of a cylindrical composite material with the special geometry. The matrix is modelling by the unit disc with different conductivity of six equal sectors. Inclusions (having different conductivity too) are symmetrically situated discs non-intersecting boundary of sectors. Mixed boundary conditions on parts of the boundary of matrix and matrix-inclusions leads to different model of composite materials. A new method to study the corresponding mathematical model is proposed. It is based on the reduction of the problem to the vector-matrix boundary value problem for analytic vectors. The method is connected with the approach by Zhorovina and Mityushev to the study of R-linear boundary value on a fan-shaped domain.