2,255 research outputs found
Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally
challenging task, especially when a large set of eigenvalues is computed, or
just when particularly large eigenvalues are sought. This is a consequence of
the highly oscillatory behaviour of the solutions corresponding to high
eigenvalues, which forces a naive integrator to take increasingly smaller
steps. We will discuss some techniques that yield uniform approximation over
the whole eigenvalue spectrum and can take large steps even for high
eigenvalues. In particular, we will focus on methods based on coefficient
approximation which replace the coefficient functions of the Sturm-Liouville
problem by simpler approximations and then solve the approximating problem. The
use of (modified) Magnus or Neumann integrators allows to extend the
coefficient approximation idea to higher order methods
Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials
According to the linear theory of elasticity, there exists a combination of different orders of stress singularity at a V-notch tip of bonded dissimilar materials. The singularity reflects a strong stress concentration near the sharp V-notches. In this paper, a new way is proposed
in order to determine the orders of singularity for two-dimensional V-notch problems. Firstly, on the basis of an asymptotic stress field in terms of radial coordinates at the V-notch tip, the governing equations of the elastic theory are transformed into an eigenvalue problem of ordinary differential equations (ODEs) with respect to the circumferential coordinate h around the notch tip. Then the interpolating matrix method established by the first author is further developed to solve the general eigenvalue problem. Hence, the singularity orders of the V-notch problem are determined through solving the corresponding
ODEs by means of the interpolating matrix method. Meanwhile, the associated eigenvectors of the displacement and stress fields near the V-notches are also obtained. These functions are essential in calculating the amplitude of the stress field described as generalized stress intensity factors of the V-notches. The present method is also available to deal with the plane V-notch problems in bonded orthotropic multi-material. Finally, numerical
examples are presented to illustrate the accuracy and the effectiveness of the method
Spectrum of a non-self-adjoint operator associated with the periodic heat equation
We study the spectrum of the linear operator subject to the
periodic boundary conditions on . We prove that the
operator is closed in with the domain in for , its spectrum consists of an infinite
sequence of isolated eigenvalues and the set of corresponding eigenfunctions is
complete. By using numerical approximations of eigenvalues and eigenfunctions,
we show that all eigenvalues are simple, located on the imaginary axis and the
angle between two subsequent eigenfunctions tends to zero for larger
eigenvalues. As a result, the complete set of linearly independent
eigenfunctions does not form a basis in .Comment: 22 pages, 10 figure
Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators
This survey is a short version of a chapter written by the first two authors
in the book [A. Henrot, editor. Shape optimization and spectral theory. Berlin:
De Gruyter, 2017] (where more details and references are given) but we have
decided here to put more emphasis on the role of the Aharonov-Bohm operators
which appear to be a useful tool coming from physics for understanding a
problem motivated either by spectral geometry or dynamics of population.
Similar questions appear also in Bose-Einstein theory. Finally some open
problems which might be of interest are mentioned.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0724
Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions
We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space , involving a linear functional but, although we seek positive solutions, the functional is not assumed to be positive for all positive . The results are new even for the classic boundary conditions of clamped or hinged ends when , because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs
- …