Finite Element Approximation of the Minimal Eigenvalue of a Nonlinear Eigenvalue Problem

© 2018, Pleiades Publishing, Ltd. The problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radiofrequency discharge at reduced pressures. A necessary and sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem is established. The original differential eigenvalue problem is approximated by the finite element method on a uniform grid. The convergence of approximate eigenvalue and approximate positive eigenfunction to exact ones is proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter

Eigenvibrations of a beam with elastically attached load

© 2016, Pleiades Publishing, Ltd.The nonlinear eigenvalue problem describing eigenvibrations of a beam with elastically attached load is investigated. The existence of an increasing sequence of positive simple eigenvalues with limit point at infinity is established. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. The problem is approximated by the finite element method with Hermite finite elements of arbitrary order. The convergence and accuracy of approximate eigenvalues and eigenfunctions are investigated

On foundations of quantum physics

Some aspects of the interpretation of quantum theory are discussed. It is emphasized that quantum theory is formulated in the Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between 'canonically conjugated' coordinate and momentum operators leads to a wrong version of quantum mechanics. The origin of time is analyzed in detail by the example of atomic collision theory. It is shown that for a closed system like the three-body (two nuclei + electron) time-dependent Schroedinger equation has no physical meaning since in the high impact energy limit it transforms into an equation with two independent time-like variables; the time appears in the stationary Schroedinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system. Following the Einstein-Rozen-Podolsky experiment and Bell's inequality the wave function is interpreted as an actual field of information in the elementary form. The relation between physics and mathematics is also discussed.Comment: This article is extended version of paper: Solov'ev, E.A.: Phys.At.Nuc. v. 72, 853 (2009

General solution of equations of motion for a classical particle in 9-dimensional Finslerian space

A Lagrangian description of a classical particle in a 9-dimensional flat Finslerian space with a cubic metric function is constructed. The general solution of equations of motion for such a particle is obtained. The Galilean law of inertia for the Finslerian space is confirmed.Comment: 10 pages, LaTeX-2e, no figures; added 2 reference

Approximation of nonlinear spectral problems in a Hilbert space

© 2015, Pleiades Publishing, Ltd. We study an eigenvalue problem with a nonlinear dependence on the parameter in a Hilbert space. We establish the existence of eigenvalues and eigenelements. The original infinite-dimensional problem is approximated by a problem in a finite-dimensional subspace. We investigate the convergence and accuracy of approximate eigenvalues and eigenelements

Eigenvibrations of a bar with elastically attached load

© 2017, Pleiades Publishing, Ltd.We study the problem on the eigenvibrations of a bar with an elastically attached load. The problem is reduced to finding the eigenvalues and eigenfunctions of an ordinary secondorder differential problem with a spectral parameter nonlinearly occurring in the boundary condition at the load attachment point. We prove the existence of countably many simple positive eigenvalues of the differential problem. The problem is approximated by a grid scheme of the finite element method. We study the convergence and accuracy of the approximate solutions

Quadrature finite element method for elliptic eigenvalue problems

© 2017, Pleiades Publishing, Ltd. A positive semi-definite eigenvalue problem for second-order self-adjoint elliptic differential operator definedon a bounded domain in the planewith smooth boundary and Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange curved triangular finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are established

Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

© 2014, Pleiades Publishing, Ltd. A condition for the existence of a minimum eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for an ordinary differential equation is determined. The problem is approximated by a mesh scheme of the finite element method. The convergence of approximate solutions to exact ones is studied. Theoretical results are illustrated by numerical experiments for a model problem

Перспективы применения внутривенного иммуноглобулина при аутоиммунных ревматических заболеваниях

The published data on efficiency of intravenous immunoglobulin (IVIG) in patients with autoimmune rheumatic diseases (RD) are analyzed. IVIG is the drug of choice in patients with Kawasaki disease, idiopathic thrombocytopenic purpura, dermatomyositis, etc. It is reasonable to use IVIG in RD patients with immune deficiency, infectious complications, ineffectiveness of glucocorticoid and cytostatic treatment in patients with developing autoimmune cytopenia, central nervous system dysfunction, and vasculitis. In patients with antiphospholipid syndrome, IVIG can be effective in case of the risk of miscarriage and ineffective anticoagulant therapy.Проанализированы данные литературы об эффективности внутривенного иммуноглобулина (ВВИГ) при аутоиммунных ревматических заболеваниях (РЗ). ВВИГ является препаратом выбора при болезни Кавасаки, идиопатической тромбоцитопенической пурпуре, дерматомиозите и др. Применение ВВИГ у больных с РЗ целесообразно при иммунодефиците, инфекционных осложнениях, неэффективности глюкокортикоидов и цитостатиков в случае развития аутоиммунной цитопении, поражении ЦНС и васкулите. При антифосфолипидном синдроме ВВИГ может быть эффективным при угрозе потери плода и отсутствии эффекта антикоагулянтов

Eigenvibrations of a beam with load

© 2017, Pleiades Publishing, Ltd. The eigenvalue problem describing eigenvibrations of a beam with load is investigated. The problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. Limit properties of eigenvalues and eigenfunctions are studied