Inverse nodal problem for Dirac operator with integral type nonlocal boundary conditions

In this paper, Dirac operator with some integral type nonlocal boundary conditions is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of some coefficients of the operator.     

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.     

Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains

We prove the well-posed solvability (in the strong sense) of complete second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients under nonlocal initial conditions. We are the first to establish the well-posed solvability of the mixed problem for the complete string vibration equation with nonstationary boundary conditions and nonlocal initial conditions.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

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.     

A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.     

Nonclassical problem with integral boundary conditions for elliptic system

In the present paper, a mixed nonclassical problem for multidimensional second-order elliptic system with Dirichlet and nonlocal integral boundary conditions is considered. Since Lax-Milgram theorem cannot be applied straightforwardly for such a nonlocal problem, we consider the problem in the spaces of vector-valued distributions with respect to one space variable with values in the spaces of functions with respect to the other space variables. We introduce special multipliers and applying them we obtain suitable new a priori estimates, and under minimal conditions on the coefficients of the elliptic operator we prove the existence and uniqueness of the solution in appropriate spaces of vector-valued distributions with values in Sobolev spaces.     

Singular Perturbation in a Nonlocal Diffusion Problem

We study a singular perturbation problem for a nonlocal evolution operator. The problem appears in the analysis of the propagation of flames in the high activation energy limit, when admitting nonlocal effects.

We obtain uniform estimates and we show that, under suitable assumptions, limits are solutions to a free boundary problem in a viscosity sense and in a pointwise sense at regular free boundary points.

We study the nonlocal problem both for a single equation and for a system of two equations.

Some of the results obtained are new even when the operator under consideration is the heat operator.     

Locally one-dimensional difference scheme for a pseudoparabolic equation with nonlocal conditions

We study a two-dimensional linear pseudoparabolic equation with nonlocal integral boundary conditions in one coordinate direction and use a locally one-dimensional method for solving this problem. We prove the stability of a finite-difference scheme based on the structure of spectrum of the difference operator with nonlocal conditions.     

On a nonlocal problem of A. A. Dezin for the Lavrent’ev-Bitsadze equation

In a special rectangular domain, for a second-order linear equation of mixed type with discontinuous coefficients and with the Lavrent’ev-Bitsadze operator in the leading part, we prove an extremum principle and existence and uniqueness theorems for the solution of a nonlocal problem stated by A.A. Dezin in his report at the Joint Soviet-American Symposium on Partial Differential Equations (Novosibirsk, 1963).     

非局部摩擦在几种塑性成形工艺中的应用

为了考虑金属材料表面微凸结构对模具与工件接触区域上的非局部摩擦效应,在几种金属塑性成形加工问题中,首次采用Oden等提出的非局部摩擦定律代替经典的库仑摩擦定律,利用主应力法或工程法建立了相应问题的积微分形式的力平衡方程．在简化的情况下,采用摄动法求得接触面上接触压力在非局部摩擦下的近似解析解,并分析了影响接触压力非局部效应的相关因素．     

Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems

This paper is concerned with an inhomogeneous nonlocal dispersal equation. We study the limit of the re-scaled problem of this nonlocal operator and prove that the solutions of the re-scaled equation converge to a solution of the Fokker-Planck equation uniformly. We then analyze the nonlocal dispersal equation of an inhomogeneous diffusion kernel and find that the heterogeneity in the classical diffusion term coincides with the inhomogeneous kernel when the scaling parameter goes to zero.     

On a problem for operator-differential second-order equations with nonlocal boundary condition

An operator-differential second-order equation with nonlocal boundary condition at zero is considered on the semiaxis. Here we give sufficient conditions on the operator coefficients for the regular solvability of the boundary-value problem. Moreover, we obtain conditions for the completeness andminimality of the derivative of the chain of eigen- and associated vectors generated by the boundary-value problem under study and establish the completeness and minimality of the decreasing elementary solutions of the operator-differential equation under consideration.     

On the index of nonlocal elliptic operators corresponding to a nonisometric diffeomorphism

We consider nonlocal elliptic operators corresponding to diffeomorphisms of smooth closed manifolds. The index of such operators is calculated. More precisely, it was shown that the index of the operator is equal to that of an elliptic boundary-value problem on the cylinder whose base is the original manifold. As an example, we study nonlocal operators on the two-dimensional Riemannian manifold corresponding to the tangential Euler operator.     

On the index instability for some nonlocal elliptic problems

The index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at the conjugation points do not affect the index of the unbounded operator. In this paper, we construct examples showing that the index may change under nonlocal perturbations with coefficients not vanishing at the points of conjugation of boundary-value conditions. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 178–193, 2007.     

On a spectral problem in the theory of the heat operator

In the theory of the heat operator, we study a spectral problem with squared spectral parameter in the boundary condition. We construct the biorthogonal system and state a nonlocal spectral problem for a complete minimal eigenfunction system.     

A nonlocal problem for the Bitsadze-Lykov equation

We study a nonlocal boundary-value problem for a degenerate hyperbolic equation. We prove that this problem is uniquely solvable if Volterra integral equations of the second kind are solvable with various values of parameters and a generalized fractional integro-differential operator with a hypergeometric Gaussian function in the kernel.     

On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator

We study a nonlocal problem for a fractional partial differential equation with the Dzhrbashyan–Nersesyan fractional differentiation operator. By separation of variables, we prove a theorem on the existence and uniqueness of a regular solution of this problem.     

Nonlocal problem for the Lavrent’ev-Bitsadze equation and its analogs in the theory of equations of mixed parabolic-hyperbolic type

We study a nonlocal interior-boundary value problem with an Erdelyi-Kober operator for the Lavrent’ev-Bitsadze equation and its analogs in the theory of equations of mixed parabolic-hyperbolic type.     

Solvability of a boundary value problem for a mixed-type equation with a partial Riemann-Liouville fractional derivative

We prove existence and uniqueness theorems for the solution of a nonlocal problem for a partial differential equation with a Riemann-Liouville fractional derivative in which the boundary condition contains a generalized fractional differentiation operator with the Gauss hypergeometric function in the kernel. We present a closed-form solution of the problem.