SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Proceedings of International Conference on Applied Analysis and Mathematical Modeling 2013.

Open Access Research

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

Allaberen Ashyralyev1 and Elif Ozturk2*

Author Affiliations

1 Department of Mathematics, Fatih University, Istanbul, 34500, Turkey

2 Department of Econometrics, Canakkale Onsekiz Mart University, Canakkale, 17200, Turkey

For all author emails, please log on.

Boundary Value Problems 2014, 2014:14  doi:10.1186/1687-2770-2014-14


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/14


Received:11 October 2013
Accepted:16 December 2013
Published:13 January 2014

© 2014 Ashyralyev and Ozturk; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this study, the Bitsadze-Samarskii type nonlocal boundary-value problem with integral condition for an elliptic differential equation in a Hilbert space H with self-adjoint positive definite operator A is considered. The second order of the accuracy difference scheme for the approximate solutions of this nonlocal boundary-value problem is presented. The well-posedness of this difference scheme in Hölder spaces with a weight is proved. The theoretical statements for the solution of this difference scheme are supported by the results of numerical example.

Keywords:
well-posedness; difference scheme; elliptic equation

1 Introduction

In 1969 Bitsadze and Samarskii [1] stated and studied a new problem in which a nonlocal condition is related to the values of the solution on parts of the boundary and on an interior curve for a uniformly elliptic equation. Furthermore, in [2-16] the Bitsadze-Samarskii type nonlocal boundary-value problems were investigated for the various differential and difference equations of elliptic type. The role played by coercive inequalities in the study of local boundary-value problems for elliptic differential equations is well known [17]. Methods of solutions of elliptic differential and difference equations have been studied extensively by many researchers (see [18-27] and the references therein). In the present paper we consider the Bitsadze-Samarskii type nonlocal boundary-value problem with integral condition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M1">View MathML</a>

(1)

for the differential equation of elliptic type in a Hilbert space H with the self-adjoint positive definite operator A with a closed domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M2">View MathML</a>. Here, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M3">View MathML</a> be a given abstract continuous function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M4">View MathML</a> with values in H, φ, and ψ are elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M6">View MathML</a> is a scalar function. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7">View MathML</a> is called a solution of problem (1) if the following conditions are satisfied:

i. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7">View MathML</a> is a twice continuously differentiable on the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M9">View MathML</a>.

ii. The element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M11">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M12">View MathML</a>, and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M13">View MathML</a> is continuous on the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M4">View MathML</a>.

iii. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M7">View MathML</a> satisfies the equation and nonlocal boundary conditions (1).

The paper is organized as follows. In Section 2 the second order of the accuracy difference scheme for the approximate solution (1) is presented. The stability, the almost coercive stability, and the coercive stability estimates for the solution of the difference scheme for an approximate solution of the nonlocal boundary-value problem with integral condition for elliptic equations are obtained. Section 3 contains the applications of Section 2. The final section is devoted to the numerical result. Theoretical statements for the solution of the second order of the accuracy difference scheme is supported by a numerical experiment.

2 The second order of the accuracy difference scheme

Let us associate the nonlocal boundary-value problem (1) with the corresponding difference problem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M16">View MathML</a>

(2)

We will study the problem (2) under the following assumption:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M17">View MathML</a>

(3)

It is well known [28] that for a self-adjoint positive definite operator A it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M18">View MathML</a> is self-adjoint positive definite and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M19">View MathML</a>, which is defined on the whole space H is a bounded operator. Here, I is the unit operator. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M20">View MathML</a>

(4)

In this paper, positive constants, which can differ in time (hence they are not a subject of precision considerations) will be indicated with M. On the other hand <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M21">View MathML</a> is used to focus on the fact that the constant depends only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M22">View MathML</a> .

Lemma 1The operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M23">View MathML</a>

has an inverse

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M24">View MathML</a>

and the following estimate is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M25">View MathML</a>

(5)

whereMdoes not depend onτ.

The proof of the estimate (5) is based on the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M26">View MathML</a>

(6)

Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M27">View MathML</a>

The estimate (6) follows from the spectral representation of A and the Cauchy inequality.

Theorem 2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M29">View MathML</a>, the solution of the problem (2) exists and the following formula holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M30">View MathML</a>

(7)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M31">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M32">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M33">View MathML</a>.

Proof

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M34">View MathML</a>

(8)

has a solution and the following formula holds [29]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M35">View MathML</a>

(9)

Applying formula (9) and the nonlocal boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M36">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M37">View MathML</a>

Since the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M38">View MathML</a>

has an inverse <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M39">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M40">View MathML</a>

Theorem 2 is proved. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M41">View MathML</a> be the linear space of the mesh functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M42">View MathML</a> with values in the Hilbert space H. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M45">View MathML</a>, Banach spaces with the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M46">View MathML</a>

The nonlocal boundary-value problem (2) is said to be stable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M47">View MathML</a> if we have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M48">View MathML</a>

Theorem 3The solutions of the difference scheme (2) under the assumption (3) satisfy the stability estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M49">View MathML</a>

(10)

Proof By [29],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M50">View MathML</a>

(11)

is proved for the solution of difference scheme (8). Then the proof of (10) is based on (11) and on the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M51">View MathML</a>

Using the formula (7) and the estimates (4), (5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M52">View MathML</a>

Theorem 3 is proved. □

Theorem 4The solutions of the difference problem (2) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M53">View MathML</a>under the assumption (3) obey the almost coercive inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M54">View MathML</a>

Proof By [29],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M55">View MathML</a>

is proved for the solution of the boundary-value problem (8). Using the estimates (4), (5) and the formula (7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M56">View MathML</a>

(12)

for the solution of difference scheme (2). Applying formula (7) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M57">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M58">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M59">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M60">View MathML</a>

(14)

To this end it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M61">View MathML</a>

(15)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M62">View MathML</a>

(16)

The estimate (15) follows from formula (13) and the estimates (4), (5). Using formula (14) and the estimates (4), (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M63">View MathML</a>

From the last estimate and the estimate (15) follows the estimate (12). Theorem 4 is proved. □

Theorem 5The difference problem (2) is well posed in the Hölder spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M44">View MathML</a>under the assumption (3) and the following coercivity inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M65">View MathML</a>

(17)

Proof By [29],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M66">View MathML</a>

(18)

is proved for the solution of difference scheme (8). Then the proof of (17) is based on (18) and on the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M67">View MathML</a>

Applying the triangle inequality, formula (7) and the estimate (15), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M68">View MathML</a>

To this end it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M69">View MathML</a>

(19)

Applying formula (14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M70">View MathML</a>

(20)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M71">View MathML</a>

Second, let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M72">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M73">View MathML</a> separately. We start with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M74">View MathML</a>. Using estimates (4), (5) and the definition of the norm of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M75">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M76">View MathML</a>

From (3) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M77">View MathML</a>

Now, let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M78">View MathML</a>. Using estimates (4), (5) and the definition of the norm of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M75">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M80">View MathML</a>

The sum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M81">View MathML</a>

is the lower Darboux integral sum for the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M82">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M83">View MathML</a>

By the lower Darboux integral sum for the integral it concludes that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M84">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M85">View MathML</a>, applying (4), (5) and the definition of the norm of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M75">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M87">View MathML</a>

The sum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M88">View MathML</a>

is the lower Darboux integral sum for the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M89">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M90">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M91">View MathML</a>

By the lower Darboux integral sum for the integral it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M92">View MathML</a>

Combining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M85">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M95">View MathML</a>

From (3) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M96">View MathML</a>

Next, let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M97">View MathML</a>. Using the estimates (4), (5), and the definition of the norm space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M98">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M99">View MathML</a>

The sum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M100">View MathML</a>

is the lower Darboux integral sum for the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M101">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M102">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M103">View MathML</a>

By the lower Darboux integral sum for the integral it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M104">View MathML</a>

Finally, let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M105">View MathML</a>. Using the estimates (4), (5), and the definition of the norm space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M98">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M107">View MathML</a>

The sum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M108">View MathML</a>

is the lower Darboux integral sum for the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M109">View MathML</a>

Thus, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M110">View MathML</a>

By the lower Darboux integral sum for the integral it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M111">View MathML</a>

Combining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M105">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M114">View MathML</a>

From (3) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M115">View MathML</a>

Combining estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M73">View MathML</a> we get the estimate (19). Theorem 5 is proved. □

3 Applications

Now, the application of Theorems 3-5 will be given. First, we consider the mixed boundary-value problem for elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M118">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M122">View MathML</a> are given sufficiently smooth functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M125">View MathML</a>. The discretization of problem (21) is carried out in two steps. In the first step, let us define the grid space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M126">View MathML</a>

We introduce the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M127">View MathML</a> of the grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M128">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M129">View MathML</a>, equipped with the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M130">View MathML</a>

To the differential operator A generated by the problem (21) we assign the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M132">View MathML</a>

(22)

acting in the space of the grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M133">View MathML</a> satisfying the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M135">View MathML</a>. It is know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> is a self-adjoint positive definite operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137">View MathML</a>. With the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> , we arrive at the nonlocal boundary-value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M139">View MathML</a>

(23)

for an infinite system of ordinary differential equations. Therefore, in the second step, equation (23) is replaced by the difference scheme (2), and we get the following difference scheme:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M140">View MathML</a>

(24)

for the numerical solution of (21).

Theorem 6Letτandhbe sufficiently small positive numbers. Then under the assumption (3), the solution of the difference scheme (24) satisfies the following stability and almost coercivity estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M141">View MathML</a>

The proof of Theorem 6 is based on Theorems 3 and 4, on the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M142">View MathML</a>

(25)

and on the symmetry properties of the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> defined by the formula (22) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137">View MathML</a>.

Theorem 7Letτand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M145">View MathML</a>be sufficiently small positive numbers. Then under the assumption (3), the solution of the difference scheme (24) satisfies the following coercivity estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M146">View MathML</a>

The proof of Theorem 7 is based on Theorem 5 and the symmetry properties of the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M147">View MathML</a> defined by formula (22).

Second, let Ω be the unit open cube in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M148">View MathML</a> with boundary S, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M149">View MathML</a>. In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M150">View MathML</a>, the Dirichlet-Bitsadze-Samarskii type mixed boundary-value problem for the multidimensional elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M151">View MathML</a>

(26)

is considered. We will study the problem (26) under the assumption (3). Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M152">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M153">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M120">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M156">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M122">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M153">View MathML</a>) are smooth functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M160">View MathML</a>. The discretization of problem (26) is carried out in two steps. In the first step let us define the grid sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M161">View MathML</a>

We introduce the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M162">View MathML</a> of the grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M163">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M164">View MathML</a>, equipped with the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M165">View MathML</a>

To the differential operator A generated by the problem (26), we assign the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M167">View MathML</a>

(27)

acting in the space of the grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M168">View MathML</a>, satisfying the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M169">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M170">View MathML</a>. It is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> is a self-adjoint positive definite operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137">View MathML</a>. With the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a>, we arrive at the nonlocal boundary-value problem for an infinite system of ordinary differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M174">View MathML</a>

(28)

In the second step, (28) is replaced by the difference scheme (2), and we get the following difference scheme:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M175">View MathML</a>

(29)

for the numerical solution of (26).

Theorem 8Letτand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M176">View MathML</a>be sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M177">View MathML</a>

The proof of Theorem 8 is based on Theorem 3 and the symmetry properties of the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> defined by (27) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137">View MathML</a>.

Theorem 9Letτand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M180">View MathML</a>be sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following almost coercivity estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M181">View MathML</a>

The proof of Theorem 9 is based on Theorem 4, on the estimate (25), on the symmetry properties of the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M131">View MathML</a> defined by (27) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137">View MathML</a>, and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137">View MathML</a>.

Theorem 10For the solutions of the elliptic difference problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M185">View MathML</a>

the following coercivity inequality holds[30]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M186">View MathML</a>

Theorem 11Letτand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M145">View MathML</a>be sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following coercivity stability estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M188">View MathML</a>

The proof of Theorem 11 is based on Theorem 5, on the symmetry properties of the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M147">View MathML</a> defined by the formula (27), and on Theorem 10 on the coercivity inequality for the solution of the elliptic difference equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M137">View MathML</a>.

4 Numerical results

We consider the Bitsadze -Samarskii type nonlocal boundary problem for the elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M191">View MathML</a>

(30)

The exact solution of this problem is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M192">View MathML</a>

In the present part for the approximate solutions of the Bitsadze-Samarskii type nonlocal boundary-value problem (30), we will use the first and second orders of the accuracy difference schemes with grid intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M194">View MathML</a> for t and x, respectively. For the approximate solution of the nonlocal boundary Bitsadze-Samarskii type problem (30), we consider the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M195">View MathML</a> of a family of grid points depending on the small parameters τ and h,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M196">View MathML</a>

Applying the first order of the accuracy difference scheme from [31] and the second order of the accuracy difference scheme (2) for the approximate solutions of the problem, we have the second-order difference equations with respect to n with matrix coefficients. To solve these difference equations, we have applied the procedure of a modified Gauss elimination method for the difference equations with respect to n with matrix coefficients. To obtain the solution of (2), we use MATLAB programming. The errors are computed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M197">View MathML</a>

of numerical solutions for different values of M and N, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M198">View MathML</a> represents the exact solution and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M199">View MathML</a> represents the numerical solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/14/mathml/M200">View MathML</a>. The results are shown in Table 1, respectively.

Table 1. The errors for first- and second-order difference scheme

5 Conclusion

In this paper, the second order of the accuracy difference scheme for the approximate solution of the Bitsadze-Samarskii type nonlocal boundary-value problem with the integral condition for elliptic equations is presented. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference scheme for elliptic equations are proved. The theoretical statements for the solution of this difference scheme are supported by the result of a numerical example. As can be seen from Table 1, the second order of the accuracy difference scheme is more accurate than the first order of the accuracy difference scheme.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

EO carried out the studies, participated in the sequence alignment and drafted the manuscript and AA carried out the studies, participated in the sequence alignment. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank Prof. Dr. PE Sobolevskii for his helpful suggestions on the improvement of this paper.

References

  1. Bitsadze, AV, Samarskii, AA: On some simplest generalizations of linear elliptic problems. Dokl. Akad. Nauk SSSR. 185(4), 739–740 (1969)

  2. Samarskii, AA: Some problems in differential equation theory. Differ. Uravn.. 16(11), 1925–1935 (1980)

  3. Sapagovas, MP: Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions. Differ. Equ.. 44(7), 1018–1028 (2008). Publisher Full Text OpenURL

  4. Ashyralyev, A: A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space. J. Math. Anal. Appl.. 344(1), 557–573 (2008). Publisher Full Text OpenURL

  5. Ashyralyev, A, Ozturk, E: The numerical solution of Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition. Abstr. Appl. Anal.. 2012, Article ID 730804 (2012)

  6. Ashyralyev, A, Ozturk, E: On a difference scheme of fourth-order of accuracy for the Bitsadze-Samarskii type nonlocal boundary value problem. Math. Methods Appl. Sci.. 36(8), 936–955 (2013). Publisher Full Text OpenURL

  7. Ashyralyev, A, Ozturk, E: On Bitsadze-Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: well-posedness. Appl. Math. Comput.. 219(3), 1093–1107 (2012). Publisher Full Text OpenURL

  8. Volkova, EA, Dosiyev, AA, Buranay, SC: On the solution of a nonlocal problem. Comput. Math. Appl.. 66(3), 330–338 (2013). Publisher Full Text OpenURL

  9. Ashyralyev, A, Tetikoğlu, FS: A note on Bitsadze-Samarskii type nonlocal boundary value problems: well-posedness. Numer. Funct. Anal. Optim.. 34(9), 939–975 (2013). Publisher Full Text OpenURL

  10. Berikelashvili, G: On a nonlocal boundary value problem for a two-dimensional elliptic equation. Comput. Methods Appl. Math.. 3(1), 35–44 (2003)

  11. Gordeziani, DG: On a method of resolution of Bitsadze-Samarskii boundary value problem. Abst. Rep. Inst. Appl. Math. Tbilisi State Univ.. 2, 38–40 (1970)

  12. Kapanadze, DV: On the Bitsadze-Samarskii nonlocal boundary value problem. Differ. Equ.. 23(3), 543–545 (1987)

  13. Il’in, VA, Moiseev, EI: Two-dimensional nonlocal boundary value problems for Poisson’s operator in differential and difference variants. Mat. Model.. 1(2), 139–159 (1990)

  14. Berikelashvili, GK: On the convergence rate of the finite-difference solution of a nonlocal boundary value problem for a second-order elliptic equation. Differ. Equ.. 39(7), 945–953 (2003)

  15. Skubaczewski, AL: Solvability of elliptic problems with Bitsadze-Samarskii boundary conditions. Differ. Uravn.. 21(4), 701–706 (1985)

  16. Ashyralyev, A, Tetikoğlu, FS: FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions. Abstr. Appl. Anal.. 2012, Article ID 454831 (2012)

  17. Ladyzhenskaya, OA, Ural’tseva, NN: Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow (1973) (Russian)

  18. Gorbachuk, VL, Gorbachuk, ML: Boundary Value Problems for Differential-Operator Equations, Naukova Dumka, Kiev (1984) (Russian)

  19. Grisvard, P: Elliptic Problems in Nonsmooth Domains, Pitman, London (1985)

  20. Agmon, S: Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton (1965)

  21. Krein, SG: Linear Differential Equations in Banach Space, Nauka, Moscow (1966) (Russian)

  22. Skubachevskii, AL: Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel (1997)

  23. Agarwal, R, Bohner, M, Shakhmurov, VB: Maximal regular boundary value problems in Banach-valued weighted spaces. Bound. Value Probl.. 18, 9–42 (2005)

  24. Sobolevskii, PE: Well-posedness of difference elliptic equations. Discrete Dyn. Nat. Soc.. 1(4), 219–231 (1997)

  25. Ashyralyev, A: Well-posed solvability of the boundary value problem for difference equations of elliptic type. Nonlinear Anal., Theory Methods Appl.. 24(2), 251–256 (1995). Publisher Full Text OpenURL

  26. Ashyralyev, A, Altay, N: A note on the well-posedness of the nonlocal boundary value problem for elliptic difference equations. Appl. Math. Comput.. 175(1), 49–60 (2006). Publisher Full Text OpenURL

  27. Agmon, S, Douglis, SA, Nirenberg, L: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math.. 17, 35–92 (1964). Publisher Full Text OpenURL

  28. Sobolevskii, PE: On elliptic equations in a Banach space. Differ. Uravn.. 4(7), 1346–1348 (Russian) (1969)

  29. Ashyralyev, A, Sobolevskii, PE: New Difference Schemes for Partial Differential Equations, Birkhäuser, Basel (2004)

  30. Sobolevskii, PE: On Difference Method for Approximate Solution of Differential Equations, Izdat. Voronezh. Gosud. Univ., Voronezh (1975)

  31. Ozturk, E: Nonlocal boundary value problems for elliptic differential and difference equations. PhD thesis, Mathematics Department, Uludag University (2013) (Turkish)