Asymptotic properties of a solution of a boundary-value problem

1970 ◽  
Vol 8 (3) ◽  
pp. 625-631 ◽  
Author(s):  
A. M. Il'in
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Asymptotic Properties

6.—On the Distribution of the Eigenvalues and the Order of the Eigenfunctions of a Fourth-order Singular Boundary Value Problem

1972 ◽  
Vol 71 (1) ◽  
pp. 61-75
Author(s):  
Jyoti Chaudhuri ◽  
W. N. Everitt
Keyword(s):  
Boundary Value Problem ◽  
Real Number ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Singular Boundary ◽  
Positive Real ◽  
The Real ◽  
Asymptotic Formulae ◽  
Eigenvalue Parameter ◽  
Image Position

SynopsisThis paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problemWith suitable restrictions placed on the real-valued coefficient q the spectrum of this problem, with respect to the eigenvalue parameter λ, is discrete; let {λn; n = 1, 2, …} and {ψn; n = 1, 2, …} be the eigenvalues and associated eigenfunctions. Asymptotic formulae are obtained for N(λ), the number of eigenvalues not exceeding the real number λ, and for ψn(x) as n→∞ where x is a fixed, positive real number.

scholarly journals PARTICULARS OF A WAVE FIELD IN A SEMI-INFINITE WAVEGUIDE WITH MIXED BOUNDARY CONDITIONS AT ITS EDGE

2021 ◽  
pp. 103-109
Author(s):  
N. Gorodetskaya ◽  
I Starovoit ◽  
T. Shcherbak
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Wave Field ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Static Problem ◽  
Special Point ◽  
Infinite Strip ◽  
Conservation Of Energy

The work is devoted to the analysis of the wave field, which is excited by the reflection of the first normal propagation Rayleigh-Lamb wave from the edge of an elastic semi-infinite strip, part of which is rigidly clamped, and part is free from stresses. The boundary value problem belongs to the class of mixed boundary value problems, the characteristic feature of which is the presence of a local feature of stresses at the point of change of the type of boundary conditions. To solve this boundary value problem, the paper proposes a method of superposition, which allows to take into account the feature of stresses due to the asymptotic properties of the unknown coefficients. Asymptotic dependences for coefficients are determined by the nature of the feature, which is known from the solution of the static problem. The criterion for the correctness of the obtained results was the control of the accuracy of the law of conservation of energy, the error of which did not exceed 2% of the energy of the incident wave for the entire considered frequency range. The paper evaluates the accuracy of the boundary conditions. It is shown that the boundary conditions are fulfilled with graphical accuracy along the entire end of the semi-infinite strip, except around a special point ($\epsilon$). In this case, along the clamped end of the semi-infinite strip in the vicinity of a special point of stress remain limited. The presence of the region $\epsilon$ and the limited stresses are due to the fact that the calculations took into account the $N$ members of the series that describe the wave field, and starting from the $N+1$ member of the series moved to asymptotic values of unknown coefficients, the number of which was also limited to $2N$. As the value $N$ increased, the accuracy of the boundary conditions increased, the region $\epsilon$ decreased, and the magnitude of the stresses near the singular point increased.

About Asymptotic and Oscillation Properties of the Dirichlet Problem for Delay Partial Differential Equations

2003 ◽  
Vol 10 (3) ◽  
pp. 495-502
Author(s):  
Alexander Domoshnitsky
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Dirichlet Boundary Value Problem ◽  
The Maximum Principle ◽  
Unbounded Solutions ◽  
All Solutions

Abstract In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet boundary value problem for the heat equation: the maximum principle is not valid, unbounded solutions appear while all solutions of the classical Dirichlet problem tend to zero at infinity, for “narrow enough zones” all solutions oscillate instead of being positive. We establish that the Dirichlet problem for the wave equation with delay can possess unbounded solutions. We estimate zones of positivity of solutions for hyperbolic equations.

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