Solution of Eigenvalue Problems for Linear Hamiltonian Systems with a Nonlinear Dependence on the Spectral Parameter

Abstract

A method for solving self-adjoint eigenproblems for linear Hamiltonian systems with equation, coefficient, and boundary conditions nonlinearly dependent on the spectral parameter is presented. The suggested approach is based on the iterative Newton procedure with spectral correction. The fast convergence of the method is demonstrated, and two-sided estimates of the eigenvalue sought are obtained. The results of the test application of the outlined algorithm are presented for the problem of the transverse natural oscillations of nonhomogeneous rods with a density defect, using the Euler–Bernoulli, Rayleigh, and Timoshenko models.