Some approximate methods of solving integral equations of mixed problems

Abstract Two algorithms are developed for investigating an integral equation (IE) that arises in the study of mixed problems of the mechanics of continuous media with boundary conditions specified on a circle. The first is a generalization of the orthogonal function method and relies on the approximate construction of the sequence of eigenvalues and a corresponding system of eigenfunctions of the integral operator of the original problem. It is shown that this approach is effective for any values of a certain non-dimensional parameter λ ϵ (0, ∞) of geometrical or physical origin, occurring in the kernel of the integral equation. The second method is applicable for small λ values and is based on Koiter's idea of approximate factorization. Its advantage is its greater accuracy compared with the previously used method, which involved approximation of the kernel of the integral equation. As an example, we present the solution of an axisymmetric problem: the impression of a stamp into an elastic half-space reinforced at the boundary by a thin cover.