General Linear Boundary Value Problems Research Articles

On a class of linear continuous-discrete systems with discrete memory

A class of linear functional differential systems with continuous and discrete times and discrete memory is considered. An explicit representation of the principal components to the general solution representation such as the fundamental matrix and the Cauchy operator is derived. The obtained representation is given in terms of the system parameters and opens a way towards efficient studying general linear boundary value problems and control problems with respect to a fixed collection of linear on-target functionals. In the study of the problems mentioned above outside the class under consideration, the systems with discrete memory can be employed as model or approximating ones. This can be useful as applied to systems with aftereffect under studying rough properties that hold under small perturbations of the parameters.

Fredholm One-Dimensional Boundary-Value Problems in Sobolev Spaces

For systems of ordinary differential equations on a compact interval, we study the character of solvability of the most general linear boundary-value problems in Sobolev spaces. We find the indices of these problems and obtain a criterion of their well-posedness.

On the construction of solutions of general linear boundary value problems for systems of functional differential equations

On the construction of solutions of general linear boundary value problems for systems of functional differential equations

Linear boundary-value problems described by Drazin invertible operators

The main subject of this paper is the study of a general linear boundary-value problem with Drazin or right Drazin (respectively, left Drazin) invertible operators corresponding to initial boundary operators. The obtained results are then employed to solve a Schro¨ dinger equation.

Linear overdetermined boundary value problems in Hilbert space

The general linear boundary value problem for an abstract functional differential equation is considered in the case that the number of boundary conditions is greater than the dimension of the null-space to the corresponding homogeneous equation. Sufficient conditions of the solvability of the problem are obtained. A case of a functional differential system with aftereffect is considered separately.

On the general linear boundary value problems for impulsive systems with singularities

Abstract. The effective sufficient conditions are established for the unique solvability of the general linear boundary value problems for impulsive systems with singularities. The singularity is understood in the sense that the matrix and vector functions of a continuous (as well as of a discrete) argument which correspond to this system may be nonsummable. Analogous results are obtained for similar impulsive systems with small parameters.

Solving boundary value problems for delay differential equations by a fixed-point method

A general linear boundary value problem for a nonlinear system of delay differential equations (DDE in short) is reduced to a fixed-point problem v=Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by piecewise linear function vh defined by its values vi=vh(ti) at grid points ti, i=0,1,…,N, where N is a given positive integer and h=max1≤i≤N(ti−ti−1). Under suitable assumptions, the existence of a fixed-point of A is equivalent to existence of so called ε(h)-approximate fixed-points of vh=Avh, which can be found by minimization of L2(n) norm of residuum vh−Avh with respect to the variables vi. These ε(h)-approximate fixed-points are used for obtaining approximate solutions of the original boundary value problem for a system of DDE. Numerical experiments with the boundary value problem for a system of delay differential equations of population dynamics as well as with two periodic boundary value problems: one for the periodic distributed delay Lotka–Volterra competition system and the second one modeling two coupled identical neurons with time-delayed connections show an efficiency of this kind of approach.

General Linear Boundary Value Problem for the Second-Order Integro-Differential Loaded Equation with Boundary Conditions Containing Both Nonlocal and Global Terms

The paper is devoted to obtaining the sufficient conditions for Fredholm property for the general boundary value problem of the second-order linear integro-differential equation. Here, the boundary conditions corresponding with the boundary value problem contain both nonlocal and global terms.

A spline-projection method for ill-posed integrodifferential equations

In this paper we consider the general linear boundary value problem for ill-posed integrodifferential equations of an arbitrarily fixed finite order. We theoretically substantiate one version of the general spline-projection method. In particular, the obtained general results allow us to deduce the convergence of the spline methods of collocation and subdomains.

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.

Approximations to the solution of boundary value problems of high order

In this paper we compare two methods for solution of a general linear boundary value problem of order 2 n. The first method is a two sided approximation method and the second method is a finite difference method. We apply these methods to a test problem for comparison.

General Linear Boundary Value Problems for Elliptic Operators withVMO Coefficients

General Linear Boundary Value Problems for Elliptic Operators withVMO Coefficients

General Linear Boundary Value Problems for Elliptic Operators with VMO Coefficients

We prove a priori estimates in Sobolev spaces for general linear elliptic boundary value problems with VMO coefficients.We give also results of existence and uniqueness of a solution.

Analyticity with respect to a parameter of a solution of a general linear boundary-value problem for a functional-differential equation

Analyticity with respect to a parameter of a solution of a general linear boundary-value problem for a functional-differential equation

Boundary value problems for singular second-order functional differential equations

We consider general linear boundary value problems for equations of the form y′ + f( x, y(τ( x)) = 0, 0 < x < 1, where τ( x) is continuous and f( x, y) has a singularity at y = 0. The results improve and extend earlier results for the case τ( x) = x, due to Taliaferro (1979) and Gatica. (1989).

Factorization and imbedding for general linear boundary value problems

A new direction in the theory of general linear boundary value problems is explored. The starting point is an explicit Volterra factorization of the Green's matrix (and related kernels) associated with the problem. This result leads to (1) imbedding of the boundary value problems, (2) initial value algorithms for their solution, and (3) comparison theorems relating two different boundary value problems with a common boundary condition. Extensions and connections with earlier work in this area are presented.

Theory of general linear boundary value problems for elliptical systems

Theory of general linear boundary value problems for elliptical systems

A LINEAR BOUNDARY VALUE PROBLEM FOR A SYSTEM OF COMPOSITE PARTIAL DIFFERENTIAL EQUATIONS

A two-variable system of first-order partial differential equations is investigated which has, in the region under consideration, one family of real characteristics and two families of imaginary characteristics. A general linear boundary value problem for the system is studied. It is proved that if a certain condition is imposed on the coefficients in the boundary conditions, there is only a finite number of linearly independent solutions of the homogeneous problem and of the adjoint homogeneous problem. A formula for the index of the above problem is derived and a necessary and sufficient condition for the solvability of the inhomogeneous problem is obtained in terms of the homogeneous adjoint problem.