In the paper a new direct numerically analytic for solution of dual integral for a wide class of boundary value problems of mathematical physics is described on an example of diffraction problems on a grating consisting of finite number of infinity thin bands lying in the same plane. It is based on authors idea to reduce dual integral equations with Fourier integrals to a singular integral equation of the first kind on a system of intervals and its consequent solution by a direct numerical method using interpolational quadratures. Both Dirichlet and Neumann boundary value problems and problems with the third and fourth boundary conditions on the bands of the grating are considered. The developed approach to the solution of diffraction problems on gratings was used successfully while constructing mathematical models of a wide class of boundary value problems in electrodynamics, radio physics and electronics.