A Nonlocal Finite-Difference Boundary-Value Problem

We investigate the stability of difference schemes for the equation of heat conduction with nonlocal boundary conditions. An example is given which in a certain sense imitates the problem with variable coefficients and has an exact solution in analytical form. It is shown that the difference operator has a simple spectrum and that multiple eigenvalues appear only in the case with constant coefficients. The simple spectrum ensures that the eigenvectors of the finite-difference problem form a basis. This enables us to apply to the nonlocal problem the theory of stability of symmetrizable difference schemes.     

Decay solution for the renewal equation with diffusion

In this paper we consider age structured equation with diffusion under nonlocal boundary condition and nonnegative initial data. We prove existence, uniqueness and the positivity of the solution to the above problem. Our main result is to get an exponential decay of the solution for large times toward such a study state. To this end we prove a weighted Poincaré–Wirtinger’s type inequality in unbounded domain.     

Incomplete comparison principle in problems about surface waves trapped by fixed and freely floating bodies

We consider problems about interaction of surface waves with fixed and freely floating bodies. We propose a new boundary value problem with integro–differential boundary conditions. This problem has physical sense only in the case of a totally submerged body, but, in the case of symmetry in the geometric or physical sense, the nonemptiness of its discrete spectrum provides the existence of eigenvalues of both problems for a fixed body and a freely floating body (in the last case, under some additional requirements which make the comparison principle incomplete). Bibliography: 25 titles. Illustrations: 5 figures.     

On the spectral stability in subspaces for difference schemes with nonlocal boundary conditions

We consider an initial-boundary value problem for the heat equation with nonlocal boundary conditions containing a parameter γ > 1. The spectrum of the main differential operator contains some number (depending on γ) of eigenvalues lying in the left complex half-plane, which results in the instability of the problem with respect to the initial data. For difference schemes approximating the original problem, we obtain a criterion for stability in the subspaces generated by stable harmonics.     

Solution of a nonlocal problem of the first kind for a hyperbolic equation

The spectral method is applied to establish solvability in the class of finite-energy generalized solutions of the mixed problem for a hyperbolic equation with a nonlocal boundary condition of the first kind. The solution is representable as a biorthogonal series in the root functions of the corresponding spectral problem. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 167–178, 2004.     

Computing eigenvalues of Sturm-Liouville problems via optimal control theory

In this paper, we shall address three problems arising in the computation of eigenvalues of Sturm-Liouville boundary value problems. We first consider a well-posed Sturm-Liouville problem with discrete and distinct spectrum. For this problem, we shall show that the eigenvalues can be computed by solving for the zeros of the boundary condition at the terminal point as a function of the eigenvalue. In the second problem, we shall consider the case where some coefficients and parameters in the differential equation are continuously adjustable. For this, the eigenvalues can be optimized with respect to these adjustable coefficients and parameters by reformulating the problem as a combined optimal control and optimal parameter selection problem. Subsequently, these optimized eigenvalues can be computed by using an existing optimal control software, MISER. The last problem extends the first to nonstandard boundary conditions such as periodic or interrelated boundary conditions. To illustrate the efficiency and the versatility of the proposed methods, several non-trivial numerical examples are included.     

Eigenvalues of Two Parameter Polynomial Operator Pencils of Waveguide Type

The spectral structure of two parameter unbounded operator pencils of waveguide type is studied. Theorems on discreteness of the spectrum for a fixed parameter are proved. Variational principles for real eigenvalues in some parts of the root zones are established. In the case of n = 1 (quadratic pencils) domains containing the spectrum are described (see Fig. 1–3). Conditions in the definition of the pencils of waveguide type arise naturally from physical problems and each of them has a physical meaning. In particular a connection between the energetic stability condition and a perturbation problem for the coefficients is given.     

On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators

We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain H ⊂ ℝ _σ ^m . A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1131 – 1136, August, 2005.     

On the spectrum of the Laplacian with frequently alternating boundary conditions

We consider a boundary problem for the Laplacian in a two-dimensional domain with frequently alternating boundary conditions. The leading terms of the asymptotic expansions of the eigenvalues and the corresponding eigenfunctions are constructed under the assumption that the limiting case is the mixed boundary problem. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 347–353, March, 1999.     

On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions

In this paper, we investigate the correct solvability for the Laplace equation with a nonlocal boundary condition in the unit ball. The considered boundary operator is of fractional order. This problem is a generalization of the well‐known Bitsadze–Samarskii problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Spectral properties of a nonlocal second-order difference operator

We consider a spectral problem for a nonlocal difference operator of second derivative with variable coefficients and with a complex parameter in the boundary condition. We study the algebraic and geometric multiplicity of the eigenvalues and the sign of their real part. We obtain conditions on the parameter which ensure that the entire spectrum of the operator lies in the right complex half-plane.     

Third-order linear differential equation with three additional conditions and formula for Green’s function

In this paper, we investigate a third-order linear differential equation with three additional conditions. We find a solution to this problem and give a formula and an existence condition for Green’s function. We compare two Green’s functions for two such problems with different additional conditions: nonlocal and classical boundary conditions. Formula applications are shown by examples.     

Green’s functions for stationary problems with nonlocal boundary conditions

In this paper, we investigate Green’s functions for various stationary problems with nonlocal boundary conditions. We express the Green’s function per Green’s function for a problem with classical boundary conditions. This property is illustrated by various examples. Properties of Green’s functions with nonlocal boundary conditions are compared with those for classical problems. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-73/09.     

A class of second‐order differential operators with eigenparameter‐dependent boundary and transmission conditions

In this study, we investigate a Sturm–Liouville type problem with eigenparameter‐dependent boundary conditions and eigenparameter‐dependent transmission conditions. By establishing a new self‐adjoint operator A associated with the problem, we construct fundamental solutions and obtain asymptotic formulae for its eigenvalues and fundamental solutions. Also we investigate some properties of its spectrum. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimates of eigenvalues of self-adjoint boundary-value problems with periodic coefficients

We consider self-adjoint boundary-value problems with discrete spectrum and coefficients periodic in a certain coordinate. We establish upper bounds for eigenvalues in terms of the eigenvalues of the corresponding problem with averaged coefficients. We illustrate the results obtained in the case of the Hill vector equation and for circular and rectangular plates with periodic coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 632–638, May, 1998.     

On the existence of a Feller semigroup with atomic measure in a nonlocal boundary condition

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region G. Its domain of definition consists of continuous functions satisfying a nonlocal condition on the boundary of the region. In general, the nonlocal term is an integral of a function over the closure of the region G with respect to a nonnegative Borel measure μ(y, dη) ∈ ∂G. It is proved that the operator is a generator of a Feller semigroup in the case where the measure is atomic. The smallness of the measure is not assumed. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 164–179.     

Hopf Bifurcation for a Maturity Structured Population Dynamic Model

This article is devoted to investigate some dynamical properties of a structured population dynamic model with random walk on (0,+∞). This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and use integrated semigroup theory to study such a partial differential equation. Moreover, a Hopf bifurcation theorem is given for this model.     

Finite element error estimates for 3D exterior incompressible flow with nonzero velocity at infinity

We consider a stationary incompressible Navier–Stokes flow in a 3D exterior domain, with nonzero velocity at infinity. In order to approximate this flow, we use the stabilized P1–P1 finite element method proposed by Rebollo (Numer Math 79:283–319, 1998). Following an approach by Guirguis and Gunzburger (Model Math Anal Numer 21:445–464, 1987), we apply this method to the Navier–Stokes system with Oseen term in a truncated exterior domain, under a pointwise boundary condition on the artificial boundary. This leads to a discrete problem whose solution approximates the exterior flow, as is shown by error estimates.     

On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions

In the paper, we study the problem on the number of zeros of eigenfunctions of the fourth-order boundary-value problem with spectral and physical parameters in the boundary conditions. We show that the number of zeros of the eigenfunctions corresponding to eigenvalues of positive type behaves in a usual way (it is equal to the serial number of an eigenvalue increased by 1), but, however, the number of zeros of the eigenfunction corresponding to an eigenvalue of negative type can be arbitrary. In the case of a sufficient smoothness of coefficients of the differential expression, we write the asymptotics in the physical parameter for such a number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 41–52, 2006.