Enhanced spatial skin–effect for free vibrations of a thick cascade junction with “super heavy” concentrated masses

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number 5N=O(ε−1) of ε-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The mass density is of order O(ε−α) on the rods from the second class and O(1) outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigen-magnitudes as ε→0, namely the case of “light” concentrated masses (a∈(0,1)), “intermediate” concentrated masses (α=1) and “heavy” concentrated masses (α∈(1,+∞")) that we divide into “slightly heavy” concentrated masses (α∈(1,2)), “moderate heavy” concentrated masses (α=2), and “super heavy” concentrated masses (alpha>2). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes in the cases α=2 and α>2. The leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions are constructed and the corresponding asymptotic estimates are proved. In addition, a new kind of high-frequency vibrations is found

Asymptotic behavior of the eigenvalues and eigenfunctions to a spectral problem in a thick cascade junction with concentrated masses

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundaryvalue problem for the Laplace operator in a thick cascade junction with concentrated masses is investigated. This cascade junction consists of the junction's body and great number 5N=O(ε−1) of ε-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The density of the junction is order O(ε−α) on the rods from the second class (the concentrated masses if α>0) and O(1) outside of them. In addition, we study the influence of the concentrated masses on the asymptotic behavior of these magnitudes in the case α=1 and α∈(0,1)

Spatial-skin effect for a spectral problem with “slightly heavy” concentrated masses in a thick cascade junction

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundary- value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number 5N=(ε−1) of ε−alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The density of the junction is of order O(ε−α) on the rods from the second class and O(1) outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigenvibrations as ε→0, namely the cases of “light” concentrated masses (α∈(0,1)), “middle” concentrated masses (α=1), “slightly heavy” concentrated masses (α∈(1,2)), “intermediate heavy” concentrated masses (α=2), and “very heavy” concentrated masses (α>2). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes if α∈(1,2)

On boundary-value problems for the laplacian in bounded and in unbounded domains with perforated boundaries

AbstractIn this paper we consider boundary-value problems in domains with perforated boundaries. We use the classification of homogenized (limit) problems depending on the ratio of small parameters, which characterize the diameter of the holes and the distance between them. We study the analogue of the Helmholtz resonator for domains with a perforated boundary

Asymptotic Approximation of Eigenelements of the Dirichlet Problem for the Laplacian in a Junction with Highly Oscillating Boundary

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a bounded domain, a part of whose boundary, depending on a small parameter $\varepsilon$, is highly oscillating; the frequency of oscillations of the boundary is of order $\varepsilon$ and the amplitude is fixed. We construct and analyze second-order asymptotic approximations, as $\varepsilon \to 0$, of the eigenelements in the case of simple eigenvalues of the limit problem

The Boundary-value Problem in Domains with Very Rapidly Oscillating Boundary

AbstractWe study the asymptotic behavior of the solution to boundary-value problem for the second order elliptic equation in the bounded domain Ωε⊂Rnwith a very rapidly oscillating locally periodic boundary. We assume that the Fourier boundary condition involving a small positive parameter ε is posed on the oscillating part of the boundary and that the (n−1)-dimensional volume of this part goes to infinity as ε→0. Under proper normalization conditions that homogenized problem is found and the estimates of the residual are obtained. Also, we construct an additional term of the asymptotics to improve the estimates of the residual. It is shown that the limiting problem can involve Dirichlet, Fourier or Neumann boundary conditions depending on the structure of the coefficient of the original problem