A.G. Sveshnikov's works in the numerical methods for the irregular waveguides design

In a series of works by Professor A.G.Sveshnikov in 1963 a mathematical model of the propagation of electromagnetic waves inside a metal waveguide with a complex shape boundary was formulated. This model is based and substantiated on the modified Galerkin method. Later it was named al the incomplete Galerkin method. The solution of the boundary value problem is presented in the form of a linear combina-tion of coordinate functions of the cross-section, the complete set of which is found from the eigenvalues-eigenvectors-problem inside the cross-section of the waveguide. The coefficients of the linear combination depend on the longitudinal coordinate. They are found from the condition of orthogonality of the discrep-ancy to the system of projection functions. In case of the Galerkin method, the set of the projection func-tions is equal to the set of coordinate functions (also cross-section depended). A boundary value problem for a system of linear ordinary differential equations of the second order is obtained as the result of the application of projection relations of orthogonality and boundary conditions of radiation in infinite re-gions. We show that the method created 60 years ago at the Department of Mathematics of the Faculty of Physics of Moscow State University is still relevant today. The increased capabilities of computers, math-ematical and programming tools make it possible to significantly expand the class of problems that are effectively solved using the incomplete Galerkin method. We present the results of solving one problem of optimizing a waveguide transformer in a loaded waveguide using an incomplete Galerkin method.