Certain properties of the roots of a weakly hyperbolic operator pencil

1989 ◽  
Vol 23 (1) ◽  
pp. 68-69 ◽  
Author(s):  
E. D. Nursultanov
Keyword(s):  
Operator Pencil ◽  
Hyperbolic Operator

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

scholarly journals Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

2021 ◽  
Vol 29 (1) ◽  
pp. 14-21
Author(s):  
Mikhail D. Malykh
Keyword(s):  
Electromagnetic Field ◽  
Differential Equations ◽  
Wave Equations ◽  
Adjoint Operator ◽  
Normal Modes ◽  
Operator Pencil ◽  
Homogeneous Case ◽  
Permittivity And Permeability ◽  
Simply Connected ◽  
Magnetic Potentials

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

Eigenfunctions of a fourth order operator pencil

2014 ◽  
Author(s):  
Abdizhahan Sarsenbi ◽  
Makhmud Sadybekov
Keyword(s):  
Fourth Order ◽  
Operator Pencil ◽  
Order Operator

scholarly journals Trapping of water waves by freely floating structures in a channel

2011 ◽  
Vol 467 (2136) ◽  
pp. 3613-3632 ◽  
Author(s):  
Sergey A. Nazarov ◽  
Juha H. Videman
Keyword(s):  
Spectral Problem ◽  
Water Waves ◽  
Variational Principles ◽  
Water Channel ◽  
Coupled System ◽  
Wave Theory ◽  
Operator Pencil ◽  
Geometric Condition ◽  
Trapped Modes ◽  
Floating Structures

This article is concerned with the existence of rigid freely floating structures capable of supporting trapped modes (time-harmonic water waves of finite energy in an unbounded domain). Under the usual assumptions of linear water-wave theory, a condition guaranteeing the existence of trapped modes is derived, and structures satisfying this geometric condition are shown to exist in a three-dimensional water channel. The sufficient condition arises from the application of variational principles to a conveniently formulated linear spectral problem, the main effort being the construction of a reduction scheme that turns the quadnic operator pencil associated with the original coupled system into a linear self-adjoint spectral problem. An example of floating bodies supporting at least four trapped modes is given.

On Properties of the Spectrum of an Operator Pencil Arising in Viscoelasticity Theory

2018 ◽  
Vol 103 (5-6) ◽  
pp. 841-845
Author(s):  
A. V. Davydov ◽  
Yu. A. Tikhonov
Keyword(s):  
Operator Pencil ◽  
Spectrum Of An Operator
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