Bitsadze Samarskii Research Articles

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.

A generalization of Bitsadze–Samarskii problem for mixed type equation

We study a boundary-value problem with Bitsadze–Samarskii conditions on boundary characteristic on a special inner curve and on a segment of degeneration of mixed type equation. Its solvability is proved by method of integral equations, and uniqueness of solution is established by means of the maximum principle.

Method of corrections by higher order differences for Poisson equation with nonlocal boundary conditions

We consider the Bitsadze–Samarskii type nonlocal boundary value problem for Poisson equation in a unit square, which is solved by a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme converges at the rate O(|h|s) in the discrete L2-norm provided that the solution of the original problem belongs to the Sobolev space with exponent s∈[2,4].

WITHDRAWN: Method of corrections by higher order differences for one nonlocal boundary-value problem

Abstract We consider the Bitsadze–Samarskii type nonlocal boundary value problem for Poisson equation in a unit square, which is solved by a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme converges at the rate O ( | h | s ) in the discrete L 2 -norm provided that the solution of the original problem belongs to the Sobolev space with exponent s ∈ [ 2 , 4 ] .

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.

On the solution of a nonlocal problem

In a rectangular domain, we consider the Bitsadze–Samarskii nonlocal boundary value problem for the two-dimensional Poisson equation. The solution of this problem is defined as a solution of the local Dirichlet boundary value problem, by constructing a special method to find a function as the boundary value on the side of the rectangle, where the nonlocal condition was given. Further, the five point approximation of the Laplace operator is used for the realization of the proposed method. Numerical experiments are illustrated in the last section to support the analysis made.

On Bitsadze–Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: Well-posedness

The well-posedness of the Bitsadze–Samarskii type nonlocal boundary value problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The stable second order of accuracy difference scheme for the approximate solution of the problem is presented. The well-posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the almost coercivity and the coercivity estimates for solutions of difference schemes for elliptic equations are obtained.

On a difference scheme of fourth order of accuracy for the Bitsadze–Samarskii type nonlocal boundary value problem

The Bitsadze–Samarskii type nonlocal boundary value problem math image for the differential equation in a Hilbert space H with the self‐adjoint positive definite operator A with a closed domain D(A) ⊂ H is considered. Here, f(t) be a given abstract continuous function defined on [0,1] with values in H, φ and ψ be the elements of D(A), and λj are the numbers from the set [0,1]. The well‐posedness of the problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The fourth order of accuracy difference scheme for approximate solution of the problem is presented. The well‐posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the stability, the almost coercivity, and the coercivity estimates for the solutions of difference schemes for elliptic equations are obtained. Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.

On the solvability of certain boundary-value problems with the Bitsadze–Samarskii condition for linear hyperbolic equations

This paper presents results on the solvability in classes of regular solutions of some boundary-value problems with Bitsadze–Samarskii condition for one-dimensional hyperbolic equations of the second order.

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze–Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.

A note on the Bitsadze–Samarskii type nonlocal boundary value problem in a Banach space

The Bitsadze–Samarskii type nonlocal boundary value problem − u ″ ( t ) + A u ( t ) = f ( t ) ( 0 ⩽ t ⩽ 1 ) , u ( 0 ) = φ , u ( 1 ) = u ( λ ) + ψ , 0 ⩽ λ < 1 , in an arbitrary Banach space E with the positive operator A is considered. The well-posedness of this boundary value problem in the spaces of smooth functions is established. The new exact Schauder's estimates of solutions of the boundary value problems for elliptic equations are obtained.

An Analog of the Bitsadze–Samarskii Problem for a Mixed Type Equation with a Fractional Derivative

An Analog of the Bitsadze–Samarskii Problem for a Mixed Type Equation with a Fractional Derivative

The Bitsadze–Samarskii Problem for a Class of Degenerate Hyperbolic Equations

The Bitsadze–Samarskii Problem for a Class of Degenerate Hyperbolic Equations

A Boundary Value Problem for a Class of Mixed Equations with the Bitsadze–Samarskii Condition on Parallel Characteristics

A Boundary Value Problem for a Class of Mixed Equations with the Bitsadze–Samarskii Condition on Parallel Characteristics