A.V. Sochilin, I.S. Eminov, S.I. Eminov

Method by Galerkin is one of general methods for numerical solution to integral equations. There is a great experience of solution to integral equations by Galerkin-s method in the theory of diffraction and in the theory of antennas. But the experience has empirical character. This work is devoted to theoretical proof of method by Galerkin. And like a base for substantiation of method by Galerkin is proved of equivalence initial equation for second type by Fredholm. Energy fields of symmetry positive operators and positive defined ones were choose as functional fields of solutions. Earlier known method by S.G. Mikhlin would hold for positive defined operators only. In this work the methods are summarized for more general case for positive operators. The fields, which included right parts of integral equations, are explicated. Effective criterion for compact integral operators is given in this work too. The explicated theory explain the great experimental experience of solution to integral equations, which was accumulated in the theory of diffraction and the theory of antennas.