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

Open Access Research

High order of accuracy difference schemes for the inverse elliptic problem with Dirichlet condition

Charyyar Ashyralyyev

Author Affiliations

Department of Mathematical Engineering, Gumushane University, Gumushane, 29100, Turkey

TAU, 2009 Street, 143, Ashgabat, 744000, Turkmenistan

Boundary Value Problems 2014, 2014:5  doi:10.1186/1687-2770-2014-5


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/5


Received:21 October 2013
Accepted:11 December 2013
Published:7 January 2014

© 2014 Ashyralyyev; 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

The overdetermination problem for elliptic differential equation with Dirichlet boundary condition is considered. The third and fourth orders of accuracy stable difference schemes for the solution of this inverse problem are presented. Stability, almost coercive stability, and coercive inequalities for the solutions of difference problems are established. As a result of the application of established abstract theorems, we get well-posedness of high order difference schemes of the inverse problem for a multidimensional elliptic equation. The theoretical statements are supported by a numerical example.

MSC: 35N25, 39A14, 39A30, 65J22.

Keywords:
difference scheme; inverse elliptic problem; high order accuracy; well-posedness; stability; almost coercive stability; coercive stability

1 Introduction

Many problems in various branches of science lead to inverse problems for partial differential equations [1-3]. Inverse problems for partial differential equations have been investigated extensively by many researchers (see [3-18] and the references therein).

Consider the inverse problem of finding a function u and an element p for the elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M1">View MathML</a>

(1.1)

in an arbitrary Hilbert space H with a self-adjoint positive definite operator A. Here, λ is a known number, φ, ξ, and ψ are given elements of H.

Existence and uniqueness theorems for problem (1.1) in a Banach space are presented in [5]. The first and second accuracy stable difference schemes for this problem have been constructed in [15]. High order of accuracy stable difference schemes for nonlocal boundary value elliptic problems are presented in [19-21].

Our aim in this work is the construction of the third and fourth order stable accuracy difference schemes for the inverse problem (1.1).

In the present paper, the third and fourth orders of accuracy difference schemes for the approximate solution of problem (1.1) are presented. Stability, almost coercive stability, and coercive stability inequalities for the solution of these difference schemes are established.

In the application, we consider the inverse problem for the multidimensional elliptic equation with Dirichlet condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M2">View MathML</a>

(1.2)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M3">View MathML</a> is the open cube in the n-dimensional Euclidean space with boundary S, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M6">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M9">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M10">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M11">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M6">View MathML</a>) are given smooth functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M14">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M15">View MathML</a>), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M17">View MathML</a> are given numbers.

The first and second orders of accuracy stable difference schemes for equation (1.2) are presented in [15]. We construct the third and fourth orders of accuracy stable difference schemes for problem (1.2).

The remainder of this paper is organized as follows. In Section 2, we present the third and fourth order difference schemes for problem (1.1) and obtain stability estimates for them. In Section 3, we construct the third and fourth order difference schemes for problem (1.2) and establish their well-posedness. In Section 4, the numerical results are given. Section 5 is our conclusion.

2 High order of accuracy difference schemes for (1.1) and stability inequalities

We use, respectively, the third and fourth order accuracy approximate formulas

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M18">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M19">View MathML</a>

(2.2)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M20">View MathML</a>. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M22">View MathML</a> is a notation for the greatest integer function. Applying formulas (2.1) and (2.2) to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M23">View MathML</a>, we get, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M24">View MathML</a>

(2.3)

the third order of accuracy difference problem and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M25">View MathML</a>

(2.4)

the fourth order of accuracy difference problem for inverse problem (1.1).

For solving of problems (2.3) and (2.4), we use the algorithm [14], which includes three stages. For finding a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M26">View MathML</a> of difference problems (2.3) and (2.4) we apply the substitution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M27">View MathML</a>

(2.5)

In the first stage, applying approximation (2.5), we get a nonlocal boundary value difference problem for obtaining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M28">View MathML</a>. In the second stage, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M29">View MathML</a> and find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M30">View MathML</a>. Then, using the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M31">View MathML</a>

(2.6)

we define an element p. In the third stage, by using approximation (2.5), we can obtain the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M32">View MathML</a> of difference problems (2.3) and (2.4). In the framework of the above mentioned algorithm for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M28">View MathML</a>, we get the following auxiliary nonlocal boundary value difference scheme:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M34">View MathML</a>

(2.7)

for the third order of accuracy difference problem (2.3) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M35">View MathML</a>

(2.8)

for the fourth order of accuracy difference problem (2.4).

For a self-adjoint positive definite operator A, it follows that [22]<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M36">View MathML</a> is a self-adjoint positive definite operator, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M38">View MathML</a>, I is the identity operator. Moreover, the bounded operator D is defined on the whole space H.

Now we give some lemmas that will be needed below.

Lemma 2.1The following estimates hold[23]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M39">View MathML</a>

Lemma 2.2The following estimate holds[23]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M40">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M41">View MathML</a>

Lemma 2.3For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M42">View MathML</a>, the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M43">View MathML</a>

has an inverse such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M44">View MathML</a>

and the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M45">View MathML</a>

(2.9)

is valid.

Proof We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M46">View MathML</a>

(2.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M47">View MathML</a>

(2.11)

Applying estimates of Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M48">View MathML</a>

(2.12)

By using the triangle inequality, formula (2.10), estimates (2.9), (2.12), and Lemma 2.2 of paper [15], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M49">View MathML</a>

for any small positive parameter τ. From that follows estimate (2.9). Lemma 2.3 is proved. □

Lemma 2.4For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M42">View MathML</a>, the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M51">View MathML</a>

has an inverse

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M52">View MathML</a>

and the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M53">View MathML</a>

(2.13)

is satisfied.

Proof We can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M54">View MathML</a>

(2.14)

where G is defined by formula (2.11) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M55">View MathML</a>

Applying estimates of Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M56">View MathML</a>

(2.15)

Using the triangle inequality, formula (2.14), estimates (2.13), (2.15), and Lemma 2.3 of paper [15], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M57">View MathML</a>

for any small positive parameter τ. From that follows estimate (2.13). Lemma 2.4 is proved. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M59">View MathML</a> be the spaces of all H-valued grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M60">View MathML</a> in the corresponding norms,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M61">View MathML</a>

Theorem 2.1Assume thatAis a self-adjoint positive definite operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M64">View MathML</a>). Then, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M65">View MathML</a>of difference problem (2.3) obeys the following stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M66">View MathML</a>

(2.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M67">View MathML</a>

(2.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M68">View MathML</a>

(2.18)

Proof We will obtain the representation formula for the solution of problem (2.7). Applying the formula [23], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M69">View MathML</a>

(2.19)

By using formula (2.19) and nonlocal boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M70">View MathML</a>

we get the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M71">View MathML</a>

(2.20)

Solving system (2.20), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M72">View MathML</a>

(2.21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M73">View MathML</a>

(2.22)

Therefore, difference problem (2.7) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M74">View MathML</a> which is defined by formulas (2.19), (2.21), and (2.22). Applying formulas (2.19), (2.21), (2.22), and the method of the monograph [23], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M75">View MathML</a>

(2.23)

The proofs of estimates (2.17), (2.18) are based on formula (2.5) and estimate (2.23). Using formula (2.5) and estimates (2.23), (2.17), we obtain inequality (2.16). Theorem 2.1 is proved. □

Theorem 2.2Suppose thatAis a self-adjoint positive definite operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M64">View MathML</a>). Then, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M65">View MathML</a>of difference problem (2.4) obeys the stability estimates (2.16), (2.17), and (2.18).

Proof By using the representation formula (2.19) for the solution of (2.8), formula (2.19), and the nonlocal boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M80">View MathML</a>

we obtain the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M81">View MathML</a>

(2.24)

Solving system (2.24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M82">View MathML</a>

(2.25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M83">View MathML</a>

(2.26)

So, the difference problem (2.8) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M74">View MathML</a>, which is defined by formulas (2.19), (2.25), and (2.26). By using formulas (2.19), (2.25), (2.26), and the method of the monograph [23], we can get the stability estimate (2.23) for the solution of difference problem (2.8). The proofs of estimates (2.17), (2.18) are based on (2.5) and (2.23). Applying formula (2.5) and estimates (2.23), (2.17), we get estimate (2.16). Theorem 2.2 is proved. □

Theorem 2.3Assume thatAis a self-adjoint positive definite operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M86">View MathML</a>. Then, the solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M87">View MathML</a>of difference problems (2.3) and (2.4) obey the following almost coercive inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M88">View MathML</a>

(2.27)

Theorem 2.4Assume thatAis a self-adjoint positive definite operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M64">View MathML</a>). Then, the solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M65">View MathML</a>of difference problems (2.3) and (2.4) obey the following coercive inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M93">View MathML</a>

(2.28)

The proofs of Theorems 2.3 and 2.4 are based on formulas (2.5), (2.19), (2.21), (2.22), (2.25), (2.26), Lemmas 2.1 and 2.2.

3 High order of accuracy difference schemes for the problem (1.2) and their well-posedness

Now, we consider problem (1.2). The differential expression [22,23]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M94">View MathML</a>

defines a self-adjoint strongly positive definite operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M95">View MathML</a> acting on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M96">View MathML</a> with the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M97">View MathML</a>

The discretization of problem (1.2) is carried out in two steps. In the first step, we define the grid spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M98">View MathML</a>

To the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M95">View MathML</a> generated by problem (1.2) we assign the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M100">View MathML</a> defined by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M101">View MathML</a>

(3.1)

acting in the space of grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M102">View MathML</a>, satisfying the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M103">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M104">View MathML</a>.

To formulate our results, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M106">View MathML</a> be spaces of the grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M107">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M108">View MathML</a>, equipped with the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M109">View MathML</a>

Applying formula (2.5) to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M100">View MathML</a>, we arrive for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M111">View MathML</a> functions, at auxiliary nonlocal boundary value problem for a system of ordinary differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M112">View MathML</a>

(3.2)

We define function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M113">View MathML</a> by formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M114">View MathML</a>

(3.3)

In the second step, auxiliary nonlocal problem (3.2) is replaced by the third order of accuracy difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M115">View MathML</a>

(3.4)

and by the fourth order of accuracy difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M116">View MathML</a>

(3.5)

Let τ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M117">View MathML</a> be sufficiently small positive numbers.

Theorem 3.1The solutions of difference schemes (3.4) and (3.5) obey the following stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M118">View MathML</a>

Theorem 3.2The solutions of difference schemes (3.4) and (3.5) obey the following almost coercive stability estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M119">View MathML</a>

Theorem 3.3The solutions of difference schemes (3.4) and (3.5) obey the following coercive stability estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M120">View MathML</a>

The proofs of Theorems 3.1-3.3 are based on the abstract Theorems 2.1-2.4, symmetry properties of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M100">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M122">View MathML</a> and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M122">View MathML</a>.

Theorem 3.4[24]

For the solution of the elliptic difference problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M124">View MathML</a>

the following coercivity inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M125">View MathML</a>

whereMdoes not depend onhand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M126">View MathML</a>.

4 Numerical results

In this section, by using the third and fourth order of the accuracy approximation, we obtain an approximate solution of the inverse problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M127">View MathML</a>

(4.1)

for the elliptic equation. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M129">View MathML</a> are the exact solutions of equation (4.1).

For the approximate solution of the nonlocal boundary value problem (3.2), consider the set of grid points

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M130">View MathML</a>

which depends on the small parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M132">View MathML</a>.

Applying approximations (3.4) and (3.5), we get, respectively, the third order of the accuracy difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M133">View MathML</a>

(4.2)

and the fourth order of the accuracy difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M134">View MathML</a>

(4.3)

for the approximate solutions of the auxiliary nonlocal boundary value problem (3.2). Applying approximation (3.3) and the second order of the accuracy in x in the approximation of A, we get the following values of the p function in the grid points:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M135">View MathML</a>

(4.4)

In this step, applying to the boundary value problem for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M136">View MathML</a> for the third and fourth order approximation in the variable t, we get, respectively, the third order of the accuracy difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M137">View MathML</a>

(4.5)

and the fourth order of the accuracy difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M138">View MathML</a>

(4.6)

We can rewrite the difference scheme (4.2) in the following matrix form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M139">View MathML</a>

(4.7)

Here, I is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140">View MathML</a> identity matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M141">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M142">View MathML</a> column matrix, A, B, C, D, E are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140">View MathML</a> square matrices. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M144">View MathML</a>

(4.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M145">View MathML</a>

(4.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M146">View MathML</a>

(4.10)

For the solution of the linear matrix equation (4.7), we use the modified Gauss elimination method [25]. Namely, we seek a solution of equation (4.7) by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M147">View MathML</a>

(4.11)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M149">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M150">View MathML</a>) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140">View MathML</a> square matrices, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M152">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M150">View MathML</a>) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M142">View MathML</a> column matrices which are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M155">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M156">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M157">View MathML</a> are the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M140">View MathML</a> zero matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M160">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M162">View MathML</a> are defined by formulas

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M163">View MathML</a>

We rewrite the difference scheme (4.5) in matrix form,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M164">View MathML</a>

(4.12)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M165">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M142">View MathML</a> column matrix, A, B, D, E are defined by formulas (4.8) and (4.9). We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M167">View MathML</a> square matrices, and C is the following matrix:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M168">View MathML</a>

(4.13)

We can write the difference scheme (4.3) in matrix form (4.12), where A, B, D, E are defined by formulas (4.8) and (4.9), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M141">View MathML</a> is defined by equation (4.10), C is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M170">View MathML</a>

We have the difference scheme (4.6) in the matrix form of equation (4.12), where A, B, D, E are defined by formulas (4.8) and (4.9), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M141">View MathML</a> is defined by formula (4.10), C is defined by equation (4.13), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M165">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M173">View MathML</a>

Now we give the results of the numerical analysis using MATLAB programs. The numerical solutions are recorded for different values of N, M; and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M174">View MathML</a> represents the numerical solutions of these difference schemes at the grid points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M175">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M176">View MathML</a> represents the numerical solutions at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M177">View MathML</a>. For comparison with the exact solutions, the errors are computed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M178">View MathML</a>

Tables 1-3 are constructed for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M182">View MathML</a>. Hence, the third order and fourth order of the accuracy difference schemes are more accurate than the second order of the accuracy difference schemes (ADS). Table 1 gives the error between the exact solution and solutions derived by difference schemes for the nonlocal problem. Table 2 includes the error between the exact p solution and approximate p derived by the difference schemes. Table 3 gives the error between the exact u solution and solutions derived by the difference schemes.

Table 1. Error<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M183">View MathML</a>

Table 2. Error<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M184">View MathML</a>

Table 3. Error<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/5/mathml/M185">View MathML</a>

5 Conclusion

In this paper, the overdetermination problem for an elliptic differential equation with Dirichlet boundary condition is considered. The third and fourth orders of accuracy difference schemes for approximate solutions of this problem are presented. Theorems on the stability, almost coercive stability, and coercive stability estimates for the solutions of difference schemes for the elliptic equation are proved. As a result of the application of established abstract theorems, we get well-posedness of high order difference schemes of the inverse problem for a multidimensional elliptic equation. Numerical experiments are given. As can be seen from Tables 1-3, the third and fourth orders of the accuracy difference schemes are more accurate than the second order of the accuracy difference scheme.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author would like to thank Prof. Allaberen Ashyralyev (Fatih University, Turkey) for his helpful suggestions in improving the quality of this paper.

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