Abstract: The modelling of physical phenomena, such as the
earth’s free oscillations, the vibration of strings, the interaction of
atomic particles, or the steady state flow in a bar give rise to Sturm-
Liouville (SL) eigenvalue problems. The boundary applications of
some systems like the convection-diffusion equation, electromagnetic
and heat transfer problems requires the combination of Dirichlet and
Neumann boundary conditions. Hence, the incorporation of Robin
boundary condition in the analyses of Sturm-Liouville problem. This
paper deals with the computation of the eigenvalues and
eigenfunction of generalized Sturm-Liouville problems with Robin
boundary condition using the finite element method. Numerical
solution of classical Sturm–Liouville problem is presented. The
results show an agreement with the exact solution. High results
precision is achieved with higher number of elements.
Abstract: This article is devoted to the numerical solution of
large-scale quadratic eigenvalue problems. Such problems arise in
a wide variety of applications, such as the dynamic analysis of
structural mechanical systems, acoustic systems, fluid mechanics,
and signal processing. We first introduce a generalized second-order
Krylov subspace based on a pair of square matrices and two initial
vectors and present a generalized second-order Arnoldi process for
constructing an orthonormal basis of the generalized second-order
Krylov subspace. Then, by using the projection technique and the
refined projection technique, we propose a restarted generalized
second-order Arnoldi method and a restarted refined generalized
second-order Arnoldi method for computing some eigenpairs of largescale
quadratic eigenvalue problems. Some theoretical results are also
presented. Some numerical examples are presented to illustrate the
effectiveness of the proposed methods.