Boundary-value problems for analytic and generalized analytic functions

This paper deals with boundary value problems of linear conjugation with shift for analytic functions in the case of piecewise continuous coefficients. Int main goal is the construction of a canonical matrix for these problems. Boundary value problems with shift for generalized analytic functions and vectors as well as differential boundary value problems are studied. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems

Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1289–1398, September, 1993.     

Method for studying boundary value problems with nonlinear boundary conditions

A constructive approach to the determination of an approximate solution of a boundary value problem with nonlinear boundary conditions g[z (0), z (T)]=0 is proposed. Existence of the exact solution is proved, and error estimates for the constructed approximate solution are provided.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 951–957, July, 1990.     

Mixed problem for x-analytical functions in a halfdisk

The mixed problem for x-analytical functions in a halfdisk with the real part of the x-analytical function defined on a part of the circle and the imaginary part defined on the rest of the circle is reduced to the problem of linear matching of p-analytical functions with different characteristics (p=x in the halfdisk and p=x/(x ² +y ² ) in the halfplane with the halfdisk cut out). The linear matching problem of p-analytical functions is analyzed and the class of p-analytical functions with the characteristic p=x/(x ² +y ² ) is shown to be equivalent to the class of x-analylical functions. An integral representation is obtained for p-analytical functions with the characteristic p=x/(x ² +y ² ) which is analogous to the standard integral representation for x-analytical functions. A procedure is developed for reducing the solution of the linear matching problem to the solution of the Fredholm integral equation of second kind.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 7–15, 1986.     

Fractional boundary value problems with integral boundary conditions

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.     

Impulsive boundary value problems with nonlinear boundary conditions

In this paper, the upper and lower solution method and Schauder’s fixed point theorem are employed in the study of boundary value problems for a class of second-order impulsive ordinary differential equations with nonlinear boundary conditions. We prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.     

Second-order boundary value problems with integral boundary conditions

In this paper we investigate the existence of positive solutions of nonlocal second-order boundary value problems with integral boundary conditions.     

Boundary-value problems for hyperbolic equations with constant coefficients

By using a metric approach, we study the problem of well-posedness of boundary-value problems for hyperbolic equations of ordern (n2) with constant coefficients in a cylindrical domain. Conditions of existence and uniqueness of solutions are formulated in number-theoretic terms. We prove a metric theorem on lower estimates of small denominators that appear when constructing solutions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 795–802, July, 1994.     

Fourth-order problems with nonlinear boundary conditions

We develop a new method of lower and upper solutions for a fourth-order nonlinear boundary value problem where the differential equation has dependence on all lower-order derivatives. Our boundary conditions are nonlinear. We will assume the functions that define the nonlinear boundary conditions are either monotone or nonmonotone. As a result we obtain existence principles which improve recent results in the literature.     

Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind

In this paper we consider two initial-boundary value problems with nonlocal conditions. The main goal is to propose a method for proving the solvability of nonlocal problems with integral conditions of the first kind. The proposed method is based on the equivalence of a nonlocal problem with an integral condition of the first kind and a nonlocal problem with an integral condition of the second kind in a special form. We prove the unique existence of generalized solutions to both problems.     

Higher order boundary value problems with nonhomogeneous boundary conditions

We study the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition

     

Stability estimates for parabolic problems with Wentzell boundary conditions

Of concern is the uniformly parabolic problem

     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.