On the solution of elliptic equations by the ray variable method

Previously unknown solutions of the inner boundary value problems for inhomogeneous linear second-order elliptic equations with sufficiently weak restrictions on the coefficients, sources, and the region shape were found. Solutions are sought in the form of a superposition of the contributions of volume and boundary sources placed on the rays arriving at the reference point from the boundaries of the region. Sources are given by means of ray variables: the direction of the ray connecting two points and the distance, measured along the ray.
The finite-analytic discretization scheme is constructed for the numerical solution of problems with discontinuous coefficients and sources. The region is divided into cells, within which the coefficients and sources are continuous, and the finite discontinuities (if any) occur at the cell boundaries. Next, the solutions are cross-linked at the boundaries. In the scheme, there is no strong dependence of the accuracy of approximation from the size and shape of the cells that inherent in finite difference schemes.