The finite electromigration boundary value problem

1994 ◽  
Vol 9 (3) ◽  
pp. 563-569 ◽  
Author(s):  
J.R. Lloyd ◽  
J. Kitchin
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Series Solution ◽  
Boundary Value ◽  
The Other ◽  
Length Effect ◽  
The Laplace Transform ◽  
Unique Approach ◽  
Finite Distance ◽  
Electromigration Lifetime

The electromigration boundary value problem is investigated for the three physically reasonable boundary conditions, assuming a perfectly blocking boundary on one side. The solution to this problem is believed to be that for nucleation dominated electromigration lifetime. The three boundary conditions investigated are the semi-infinite constant vacancy source of Shatzkes and Lloyd,9 the closed system of De Groot13 and Kirchheim and Käber,19 and the heretofore unsolved constant vacancy source at a finite distance from the blocking boundary. It is argued that the first is unrealistic in that there is no length effect possible, which has been repeatedly observed experimentally. The second is argued to be too restrictive to account for failure, leaving the last as the most physically reasonable under most circumstances. The deceptively simple appearance of the boundary conditions belies a complex, double infinite series solution arrived at by a unique approach to inverting the Laplace transform of the solution. The solution correctly predicts the experimental observations of a length effect and, combined with the understanding provided by the solutions under the other two boundary conditions, the effect of a thick passivation layer on electromigration lifetime.

Uniqueness theorem for Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions

2016 ◽  
Vol 11 (2) ◽  
pp. 167-170
Author(s):  
A.M. Akhtyamov ◽  
Kh.R. Mamedov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Frequency Spectrum ◽  
Natural Frequencies ◽  
Boundary Value ◽  
Initial Boundary ◽  
The Other ◽  
Elasticity Coefficient ◽  
Nonseparated Boundary Conditions ◽  
String Vibrations

Consider the string, which vibrates in a medium with the variable elasticity coefficient q(x). Interesting to follow the inverse problem: is it possible to determine the variable elasticity coefficient q(x) by the natural frequencies of string vibrations. In 1946, G. Borg has been shown that a spectrum of frequencies is not sufficient to uniquely identify the medium elasticity coefficient q(x). He offered the use of two frequency spectrum to uniquely identify of the medium elasticity coefficient q(x). The second frequency spectrum is obtained by fastening the string to change at one of its ends to the other fastening. It was shown that these two frequency spectra already sufficient to uniquely identify q(x) and the boundary conditions of both problems. The case where the string fastening at one end depends on the other end fastening, is more difficult to solve. The boundary conditions, appropriate for the occasion, called nonseparated. Two spectra (of two boundary value problems) to restore both q(x), and the nonseparated boundary conditions are not enough. In modern studies the spectra of the two eigenvalues boundary problems and an infinite sequence of signs is generally used for an uniqueness recovery. While this approach is useful in theoretical mathematics, it is inconvenient for the mechanics, because not clear the physical meaning of the corresponding sequence of signs. In this article, instead of the two spectra and the sequence of signs as the spectral data are offered to use 7 of the eigenvalues of the initial boundary value problem, the spectrum, and the so-called norming constants of other boundary value problem. The physical sense of these data is quite clear. The first 7 eigenvalues of an initial boundary problem mean the first 7 natural frequencies of string vibrations. Norming constants represent norms from eigenfunctions. The spectrum and norming constants express a so-called spectral function. The spectral function gives a frequency spectrum with columns of vibrations amplitudes characteristics for string vibrations with other types of fastening.

scholarly journals Solution of Nonlinear High Order Multi-Point Boundary Value Problems By Semi-Analytic Technique

2014 ◽  
Vol 11 (2) ◽  
pp. 220-228
Author(s):  
Baghdad Science Journal
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Series Solution ◽  
Boundary Value ◽  
Convergent Series ◽  
High Order ◽  
Point Boundary ◽  
Physical Problems ◽  
Nonlinear High Order

In this paper, we present new algorithm for the solution of the nonlinear high order multi-point boundary value problem with suitable multi boundary conditions. The algorithm is based on the semi-analytic technique and the solutions are calculated in the form of a rapid convergent series. It is observed that the method gives more realistic series solution that converges very rapidly in physical problems. Illustrative examples are provided to demonstrate the efficiency and simplicity of the proposed method in solving this type of multi- point boundary value problems.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

Sign in / Sign up

Export Citation Format

Share Document