scholarly journals On the asymptotic behavior of the spectrum of a sixth-order differential operator, whose potential is the delta function

2020 ◽  
Vol 22 (3) ◽  
pp. 280-305
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Delta Function ◽  
Discontinuous Coefficients ◽  
Sixth Order ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Order Differential Operator ◽  
Indicator Diagram ◽  
Dirac Delta

In this paper we propose a new method for studying differential operators with discontinuous coefficients.We consider a sequence of sixth-order differential operators with piecewise-smooth coefficients. The limit of the sequence of these operators’ potentials is the Dirac delta function. The boundary conditions are separated. To correctly determine solutions of differential equations with discontinuous coefficients at the points of discontinuity, “gluing” conditions are required. Asymptotic solutions were written out for large values of the spectral parameter, with the help of them the “gluing” conditions were studied and the boundary conditions were investigated. As a result, we derive an eigenvalues equation for the operator under study, which is an entire function. The indicator diagram of the eigenvalues equation, which is a regular hexagon, is investigated. In various sectors of the indicator diagram, the method of successive approximations has been used to find the eigenvalues asymptotics of the studied differential operators. The limit of the asymptotic of the spectrum determines the spectrum of the sixth-order operator, whose potential is the delta function.

scholarly journals Asymptotics of the spectrum of even-order differential operators with discontinuos weight functions

2020 ◽  
Vol 22 (1) ◽  
pp. 48-70
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Boundary Value ◽  
Weight Functions ◽  
Order Differential Operator ◽  
Eighth Order ◽  
Entire Family ◽  
Indicator Diagram

The boundary-value problem for an eighth-order differential operator whose potential is a piecewise continuous function on the segment of the operator definition is studied. The weight function is piecewise constant. At the discontinuity points of the operator coefficients, the conditions of "conjugation" must be satislied which follow from physical considerations. The boundary conditions of the studied boundary value problem are separated and depend on several parameters. Thus, we simultaneously study the spectral properties of entire family of differential operators with discontinuous coefficients. The asymptotic behavior of the solutions of differential equations defining the operator is obtained for large values of the spectral parameter. Using these asymptotic expansions, the conditions of "conjugation" are investigated; as a result, the boundary conditions are studied. The equation on eigenvalues of the investigated boundary value problem is obtained. It is shown that the eigenvalues are the roots of some entire function. The indicator diagram of the eigenvalue equation is investigated. The asymptotic behavior of the eigenvalues in various sectors of the indicator diagram is found.

scholarly journals Application of the successive approximation method to the analysis of bending plates

2018 ◽  
Vol 251 ◽  
pp. 04058
Author(s):  
Radek Gabbasov ◽  
Vladimir Filatov ◽  
Nikita Ryasny
Keyword(s):  
Boundary Conditions ◽  
Successive Approximation ◽  
Approximation Method ◽  
High Accuracy ◽  
Numerical Results ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Successive Approximation Method ◽  
Medium Thickness ◽  
Bending Plates

This work presents an algorithm for calculating the bending plates of medium thickness according to the Reissner’. To obtain numerical results, the method of successive approximations (MSA) is used. This method has high accuracy and fast convergence, which was confirmed by the solution of a range of tasks. Publication of the results of the calculation of plates of medium thickness with the boundary conditions revised here is supposed to be in the following articles.

scholarly journals THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS

World Science ◽  
2019 ◽  
Vol 1 (11(51)) ◽  
pp. 31-39
Author(s):  
Zelensky A. G.
Keyword(s):  
Boundary Conditions ◽  
High Efficiency ◽  
Boundary Value ◽  
Three Dimensional ◽  
Median Plane ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Shallow Shells ◽  
Common Solution ◽  
Arbitrary Thickness

The method of sequential approximations (MSA) in mathematical theory (MT) of transversal-isotropic shallow shells of arbitrary thickness is developed. MT takes into account all components of stress-strain state (SSS). SSS and boundary conditions are considered to be functions of three varia-bles. Three-dimensional problems are reduced to two- dimensional decompositions of all the compo-nents of the SSS into series in the transverse coordinate using Legendre polynomials and using the Reisner variational principle. The boundary conditions for stresses on the front surfaces of the shell are fulfilled precisely. Previous studies have shown the high efficiency of this MT. The boundary-value problem for a shallow shell is reduced to sequences of two boundary-value problems for the respective plates. One sequence describes symmetric deformation relative to the median plane, and the other sequence is skew symmetric. MSA makes it easier to find a common solution of differential equations (DE) for shallow shells. Highly accurate results for SSS are already in the first approxi-mation. MSA can be used when solving problems for shallow shells by other theories.

Stresses in a Stretched Slab Having a Spherical Cavity

1959 ◽  
Vol 26 (2) ◽  
pp. 235-240
Author(s):  
Chih-Bing Ling
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Spherical Cavity ◽  
Analytic Solution ◽  
Linear Equations ◽  
Hankel Transform ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Biharmonic Functions ◽  
Infinite Slab

Abstract This paper presents an analytic solution for an infinite slab having a symmetrically located spherical cavity when it is stretched by an all-round tension. The required stress function is constructed by combining linearly two sets of periodic biharmonic functions and a biharmonic integral. The sets of biharmonic functions are derived from two fundamental functions specially built up for the purpose. The arbitrary functions involved in the biharmonic integral are first adjusted to satisfy the boundary conditions on the surfaces of the slab by applying the Hankel transform of zero order. Then the stress function is expanded in spherical co-ordinates and the boundary conditions on the surface of the cavity are satisfied by adjusting the coefficients of superposition attached to the sets of biharmonic functions. The resulting system of linear equations is solved by the method of successive approximations. The solution is finally illustrated by numerical examples for two radii of the cavity.

Point interactions: boundary conditions or potentials with the Dirac delta function

2010 ◽  
Vol 88 (11) ◽  
pp. 809-815 ◽  
Author(s):  
Salvatore De Vincenzo ◽  
Carlet Sánchez
Keyword(s):  
Boundary Conditions ◽  
Singular Potential ◽  
Quantum Particle ◽  
Delta Function ◽  
Point Interaction ◽  
Point Interactions ◽  
Free System ◽  
Dirac Delta ◽  
Singular Interaction ◽  
Zero Range

We study the problem of a nonrelativistic quantum particle moving on a real line with an idealized and localized singular interaction with zero range at x = 0 (i.e., a point interaction there). This kind of system can be described in two ways: (i) by considering an alternative free system (i.e., without the singular potential) but excluding the point x = 0 (In this case, the point interaction is exclusively encoded in the boundary conditions.) and (ii) by explicitly considering the singular interaction by means of a local singular potential. In this paper we relate, compare, and discuss, in a simple and pedagogical way these two equivalent approaches. Our main goal in this paper is to introduce the essential ideas about point interactions in a very accesible form to advanced undergraduates.

scholarly journals Приближенные граничные условия для задачи нахождения оптических коэффициентов ультратонких металлических пленок в СВЧ и ТГЦ диапазонах

2020 ◽  
Vol 128 (9) ◽  
pp. 1327
Author(s):  
П.С. Глазунов ◽  
В.А. Вдовин ◽  
В.Г. Андреев
Keyword(s):  
Boundary Conditions ◽  
Maximum Error ◽  
Dielectric Substrate ◽  
Thickness Dependence ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Transmission Coefficients ◽  
Fabry Perot ◽  
Approximate Boundary Conditions ◽  
Analytical Expressions

Approximate boundary conditions for a problem of calculating the optical coefficients of a system composed of inhomogeneous ultrathin metallic film with an arbitrary thickness dependence of conductivity deposited on dielectric substrate are obtained. The derivation of the boundary conditions is based on the Picard method of successive approximations. Analytical expressions for the errors in calculating the optical coefficients with use of the proposed approximate boundary conditions are presented. It is shown that the error increases with the frequency and the film thickness increasing. The maximum error for films of 10 nm-thickness does not exceed 10.7% at 1 THz. As an example, the complex optical coefficients of a system similar to Fabry-Perot etalon and a metal film without a substrate with model thickness dependence of conductivity are calculated. The coincidence between the results of numerical simulation and calculations performed with approximate boundary conditions is shown. The possibility of direct calculating the average conductivity of a film from experimentally measured reflection and transmission coefficients is demonstrated.

scholarly journals On the crossings number of a hyperplane by a stable random process

2018 ◽  
Vol 10 (2) ◽  
pp. 346-351
Author(s):  
M.M. Osypchuk
Keyword(s):  
Fractional Derivative ◽  
Random Process ◽  
Stable Process ◽  
Delta Function ◽  
Successive Approximations ◽  
Characteristic Functions ◽  
Method Of Successive Approximations ◽  
Bounded Solutions ◽  
Discrete Approximations ◽  
Derivative Operator

The numbers of crossings of a hyperplane by discrete approximations for trajectories of an $\alpha$-stable random process (with $1<\alpha<2$) and some processes related to it are investigated. We consider an $\alpha$-stable process is killed with some intensity on the hyperplane and a pseudo-process that is formed from the $\alpha$-stable process using its perturbation by a fractional derivative operator with a multiplier like a delta-function on the hyperplane. In each of these cases, the limit distribution of the crossing number of the hyperplane by some discret approximation of the process is related to the distribution of its local time on this hyperplane. Integral equations for characteristic functions of these distributions are constructed. Unique bounded solutions of these equations can be constructed by the method of successive approximations.

scholarly journals Sixth order differential operators with eigenvalue dependent boundary conditions

2013 ◽  
Vol 7 (2) ◽  
pp. 378-389 ◽  
Author(s):  
Manfred Möller ◽  
Bertin Zinsou
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Operator Pencil ◽  
Sixth Order ◽  
Finite Multiplicity ◽  
Quadratic Operator ◽  
Upper Half Plane ◽  
Eigenvalue Dependent Boundary Conditions

We consider eigenvalue problems for sixth-order ordinary differential equations. Such differential equations occur in mathematical models of vibrations of curved arches. With suitably chosen eigenvalue dependent boundary conditions, the problem is realized by a quadratic operator pencil. It is shown that the operators in this pencil are self-adjoint, and that the spectrum of the pencil consists of eigenvalues of finite multiplicity in the closed upper half-plane, except for finitely many eigenvalues on the negative imaginary axis.

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