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Spectral Problem (spectral + problem)

Distribution by Scientific Domains
Mathematics and Statistics66%

Selected Abstracts

Spectral problems for operator matrices

MATHEMATISCHE NACHRICHTEN, Issue 12-13 2005
A. Bátkai
Abstract We study spectral properties of 2 × 2 block operator matrices whose entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components. We investigate closability in the product space, essential spectra and generation of holomorphic semigroups. Application is given to several models governed by ordinary and partial differential equations, for example containing delays, floating singularities or eigenvalue dependent boundary conditions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Eigenvalue analysis of temperature distribution in composite walls

INTERNATIONAL JOURNAL OF ENERGY RESEARCH, Issue 13 2001
Galip Oturanç
Abstract The transient heat conduction problem in two-layer composite wall is solved analytically using spectral analysis. Eigenvalues and corresponding eigenfunctions of the spectral problem for the temperature distribution in composite walls are analysed using the Rouche Theorem. The number of eigenvalues is obtained and the temperature distribution of this complicated problem is given by a formula with calculated eigenvalues. The analytical solution obtained is in explicit form and provides easy determination of temperature rise in heating and thawing applications of composite materials. Copyright © 2001 John Wiley & Sons, Ltd. [source]

The VMFCI method: A flexible tool for solving the molecular vibration problem

JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 5 2006
P. Cassam-Chenaď
Abstract The present article introduces a general variational scheme to find approximate solutions of the spectral problem for the molecular vibration Hamiltonian. It is called the "vibrational mean field configuration interaction" (VMFCI) method, and consists in performing vibrational configuration interactions (VCI) for selected modes in the mean field of the others. The same partition of modes can be iterated until self-consistency, generalizing the vibrational self-consistent field (VSCF) method. As in contracted-mode methods, a hierarchy of partitions can be built to ultimately contract all the modes together. So, the VMFCI method extends the traditional variational approaches and can be included in existing vibrational codes based on the latter approaches. The flexibility and efficiency of this new method are demonstrated on several molecules of atmospheric interest. © 2006 Wiley Periodicals, Inc. J Comput Chem 27: 627,640, 2006 [source]

On the vibrations of a plate with a concentrated mass and very small thickness

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2003
D. Gómez
Abstract We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ,. The density is of order O(,,m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ,,O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m: Low- and high-frequency vibrations are studied for m>2. We also consider the case where the thickness plate h depends on ,; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Genesis of solitons arising from individual Flows of the Camassa-Holm hierarchy

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2006
Enrique Loubet
The present work offers a detailed account of the large-time development of the velocity profile run by a single "individual" Hamiltonian flow of the Camassa-Holm (CH) hierarchy, the Hamiltonian employed being the reciprocal of any eigenvalue of the underlying spectral problem. In this simpler scenario, I prove some of the conjectures raised by McKean [27]. Notably, I confirm the ultimate shaping into solitons of the cusps that appear, near blowup sites, of any velocity profile emanating from an initial disposition for which breakdown of the wave in finite time is sure to happen. The careful large-time asymptotic analysis is carried from exact expressions describing the velocity in terms of initial data, the integration involving a "Lagrangian" scale and three "theta functions," the rates at which the latter reach their common values at each end of the line characterizing the region where soliton genesis is expected. In fact, the present method also suggests how solitons may arise from initial conditions not leading to breakdown. The full CH flow is nothing but a superposition of such commuting "individual" actions. Therein lies the hope that the present account will pave the way to elucidate soliton formation for more complex flows, in particular for the CH flow itself. © 2005 Wiley Periodicals, Inc. [source]

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2005
Liliana Borcea
We consider finite difference approximations of solutions of inverse Sturm-Liouville problems in bounded intervals. Using three-point finite difference schemes, we discretize the equations on so-called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm-Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand-Levitan formulation. © 2005 Wiley Periodicals, Inc. [source]