Type Nonlocal Boundary Value Problem Research Articles

On The Stability of Nonlocal Boundary Value Problem for Schrödinger-Parabolic Equations

In the present article, a problem for a Schrödinger-parabolic equation with nonlocal boundary value condition is considered. The stability estimates are established for the solution of Schrödinger-parabolic problem. In applications, the stability estimates of mixed type nonlocal boundary value problem for Schrödinger-parabolic equations are constructed. On the other hand, an example is considered and some error results of numerical experiments are presented in order to verify theoretical statements.

A remark on elliptic differential equations on manifold

For elliptic boundary value problems of nonlocal type in Euclidean space, the well posedness has been studied by several authors and it has been well understood. On the other hand, such kind of problems on manifolds have not been studied yet. Present article considers differential equations on smooth closed manifolds. It establishes the well posedness of nonlocal boundary value problems of elliptic type, namely Neumann-Bitsadze-Samarskii type nonlocal boundary value problem on manifolds and also DirichletBitsadze-Samarskii type nonlocal boundary value problem on manifolds, in H¨older spaces. In addition, in various H¨older norms, it establishes new coercivity inequalities for solutions of such elliptic nonlocal type boundary value problems on smooth manifolds.

Well-Posedness of a Fourth Order of Accuracy Difference Scheme for Bitsadze–Samarskii-Type Problem

ABSTRACTIn the present study, a fourth order of accuracy difference scheme for the approximate solution of the Bitsadze–Samarskii type nonlocal boundary value problem with the integral condition is investigated. Theorem on well-posedness of the difference scheme in the difference analogue of Hölder spaces with a weight is established. In applications, coercive stability estimates for the solutions of difference schemes of nonlocal boundary value problems for elliptic problems are obtained.

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 ] .

Stability of difference schemes for Bitsadze-Samarskii type nonlocal boundary value problem involving integral condition

In this study, the stable difference schemes for the numerical solution of Bitsadze-Samarskii type nonlocal boundary-value problem involving integral condition for the elliptic equations are studied. The second and fourth orders of the accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples.

A Note on Bitsadze-Samarskii Type Nonlocal Boundary Value Problems: Well-Posedness

The Bitsadze-Samarskii type nonlocal boundary value problem for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The well posedness of this problem in Hölder spaces without a weight is established. The coercivity inequalities for the solution of the Neumann-Bitsadze-Samarskii type nonlocal boundary value problem for elliptic equations are obtained. The first order of accuracy difference scheme for the approximate solution of this problem is presented. The well posedness of this difference scheme in difference analogue of Hölder spaces is established. In applications, the stability, the almost coercivity and the coercivity estimates for solutions of difference schemes for elliptic equations are obtained.

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.

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.