Determination of the frequencies and forms of oscillations of non-uniform distributed systems with boundary conditions of the third kind

Abstract

An effective method of constructing a solution of the eigenvalue and eigenfunction problem for an essentially non-uniform oscillatory system with variable distributed parameters is developed. The proposed approach is based on a combination of a variational approach, the theory of boundary-value problems and perturbation methods. An original determination of the small parameter of the problem is given and a recurrence algorithm for the successive refinement of the eigenvalues and eigenfunctions is proposed, which leads to accelerated (quadratic) convergence of the Newton's “method of tangents” type. A commentary is made on the method and is illustrated by the solution of model examples.