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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 548017, 11 pages
http://dx.doi.org/10.1155/2013/548017
Research Article

Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

1Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey
2TAU, Gerogly Street 143, 74400 Ashgabat, Turkmenistan

Received 5 July 2013; Revised 14 August 2013; Accepted 20 August 2013

Academic Editor: Abdullah Said Erdogan

Copyright © 2013 Charyyar Ashyralyyev and Mutlu Dedeturk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A …finite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the fi…rst and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem.

1. Introduction

It is well known that inverse problems arise in various branches of science (see [1, 2]). The theory and applications of well-posedness of inverse problems for partial differential equations have been studied extensively by many researchers (see, e.g., [317] and the references therein). One of the effective approaches for solving inverse problem is reduction to nonlocal boundary value problem (see, e.g., [6, 8, 11]). Well-posedness of the nonlocal boundary value problems of elliptic type equations was investigated in [1825] (see also the references therein).

In [4], Orlovsky proved existence and uniqueness theorems for the inverse problem of finding a function and an element for the elliptic equation in an arbitrary Hilbert space with the self-adjoint positive definite operator A:

In [11], the authors established stability estimates for this problem and studied inverse problem for multidimensional elliptic equation with overdetermination in which the Dirichlet condition is required on the boundary.

In present work, we study inverse problem for multidimensional elliptic equation with Dirichlet-Neumann boundary conditions.

Let be the open cube in the -dimensional Euclidean space with boundary and . In , we consider the inverse problem of finding functions and for the multidimensional elliptic equation

Here, and are known numbers,   ,   ,  ,    , and    are given smooth functions, and also     .

The aim of this paper is to investigate inverse problem (2) for multidimensional elliptic equation with Dirichlet-Neumann boundary conditions. We obtain well-posedness of problem (2). For the approximate solution of problem (2), we construct first and second order of accuracy in and difference schemes with second order of accuracy in space variables. Stability and coercive stability estimates for these difference schemes are established by applying operator approach. The modified Gauss elimination method is applied for solving these difference schemes in a two-dimensional case.

The remainder of this paper is organized as follows. In Section 2, we obtain stability and coercive stability estimates for problem (2). In Section 3, we construct the difference schemes for (2) and establish their well-posedness. In Section 4, the numerical results in a two-dimensional case are presented. Section 5 is conclusion.

2. Well-Posedness of Inverse Problem with Overdetermination

It is known that the differential expression [26] defines a self-adjoint positive definite operator acting on with the domain .

Let be the Hilbert space . By using abstract Theorems 2.1 and 2.2 of paper [11], we get the following theorems about well-posedness of problem (2).

Theorem 1. Assume that is defined by formula (3), . Then, for the solutions of inverse boundary value problem (2), the stability estimates are satisfied: where is independent of , and .
Here, is the space obtained by completion of the space of all smooth -valued functions on with the norm

Theorem 2. Assume that is defined by formula (3), . Then, for the solution of inverse boundary value problem (2), coercive stability estimate holds, where is independent of , and .

3. Difference Schemes and Their Well-Posedness

Suppose that is defined by formula (3). Then (see [26]), is a self-adjoint positive definite operator and which is defined on the whole space is a bounded operator. Here, is the identity operator.

Now we present the following lemmas, which will be used later.

Lemma 3. The following estimates are satisfied (see [27]):

Lemma 4. For and for the operator , the operator has an inverse and the estimate is satisfied, where does not depend on .
Proof of Lemma 4 is based on Lemma 3 and representation

Lemma 5. For and for the operator the operator has an inverse and the estimate is valid, where is independent of .

Proof. We have that where
By using estimates of Lemma 3, we have that where is independent of . Using the triangle inequality, formula (13), and estimates (8) and (15), we obtain for sufficiently small positive . From that it follows estimate (11). Lemma 5 is proved.

Further, we discretize problem (2) in two steps. In the first step, we define the grid spaces

Introduce the Hilbert space and of grid functions , defined on , equipped with the norms respectively.

To the differential operator generated by problem (2) we assign the difference operator defined by formula (3), acting in the space of grid functions , satisfying the condition for all . Here, is an approximation of .

By using , for obtaining functions, we arrive at problem For finding a solution of problem (19) we apply the substitution where is the solution of nonlocal boundary value problem; a system of ordinary differential equations and unknown function is defined by formula

Thus, we consider the algorithm for solving problem (19) which includes three stages. In the first stage, we get the nonlocal boundary value problem (21) and obtain . In the second stage, we put and find . Then, using (22), we obtain . Finally, in the third stage, we use formula (20) for obtaining the solution of problem (19).

In the second step, we approximate (19) in variable . Let be the uniform grid space with step size , where is a fixed positive integer. Applying the approximate formulas for , problem (19) is replaced by first order of accuracy difference scheme and second order of accuracy difference scheme For approximate solution of nonlocal problem (21), we have first order of accuracy difference scheme and second order of accuracy difference scheme respectively.

Theorem 6. Let and be sufficiently small positive numbers. Then, for the solutions of difference schemes (24) and (25) the stability estimates hold, where is independent of ,  , and .

Theorem 7. Let and be sufficiently small positive numbers. Then, for the solutions of difference schemes (24) and (25) the following almost coercive stability estimate holds, where is independent of ,  , and .

Proofs of Theorems 6 and 7 are based on the symmetry property of operator , on Lemmas 35, the formulas for difference scheme (24), for difference scheme (25), and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .

Theorem 8 (see [28]). For the solution of the elliptic difference problem the following coercivity inequality holds: where does not depend on and .

4. Numerical Results

We have not been able to obtain a sharp estimate for the constants figuring in the stability estimates. Therefore, we will give the following results of numerical experiments of the inverse problem for the two-dimensional elliptic equation with Dirichlet-Neumann boundary conditions It is clear that and are the exact solutions of (34).

We can obtain by formula , where is the solution of the nonlocal boundary value problem and is the solution of the boundary value problem

Introduce small parameters and such that . For approximate solution of nonlocal boundary value problem (35) we consider the set of a family of grid points

Applying (21), we obtain difference schemes of the first order of accuracy in and the second order of accuracy in for the approximate solutions of the nonlocal boundary value problem (35), and for the approximate solutions of the boundary value problem (36).

By using (22) and second order of accuracy in approximation of , we get the following values of in grid points:

We can rewrite difference scheme (38) in the matrix form Here, is the identity matrix, , , are square matrices, and is a column matrix which are defined by

For solving (41) we use the modified Gauss elimination method (see [29]). Namely, we seek solution of (41) by the formula where ,   are square matrices and    are column matrices. For , we get formulas where is the identity matrix and is the zero column vector.

Futher, we rewrite difference scheme (39) in the matrix form

Here, and are defined by (42) and (43) and column matrix is defined by

Now we present second order of accuracy in and difference schemes for problems (35) and (36). Applying (27) and formulas for sufficiently smooth function we get difference scheme for nonlocal problem (35), and difference scheme for boundary value problem (36).

By difference scheme (52), we write in matrix form where are defined by (42), (43), (44), and is defined by

We seek solution of (54) by the formula where    are square matrices and are column matrices. For the solution of difference equation (41) we need to use the following formulas for : where and are the zero column vector. For and we have where

We can rewrite difference scheme (53) in the matrix form where are defined by (42), (49), (43), and (44) and is defined by

Now, we give the results of the numerical realization of finite difference method for (34) by using MATLAB programs. The numerical solutions are recorded for and different values of . Grid functions , represent the numerical solutions of difference schemes for auxiliary nonlocal problem (35) and inverse problem (34) at , respectively. Grid function calculated by (40) represents numerical solution at for unknown function . The errors are computed by the norms

Tables 13 present the error between the exact solution and numerical solutions derived by corresponding difference schemes. The results are recorded for and 160, respectively. The tables show that the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme for both auxiliary nonlocal and inverse problems. Table 1 contains error between the exact and approximate solutions of auxiliary nonlocal boundary value problem (35). Table 2 includes error between the exact and approximate solutions of inverse problem (34). Table 3 represents error between the exact solution of inverse problem (34) and approximate solution which is derived by the first and second orders accuracy of difference schemes.

tab1
Table 1: Error analysis for nonlocal problem.
tab2
Table 2: Error analysis for p.
tab3
Table 3: Error analysis for u.

5. Conclusion

In this paper, inverse problem for multidimensional elliptic equation with Dirichlet-Neumann conditions is considered. The stability and coercive stability estimates for solution of this problem are established. First and second order of accuracy difference schemes are presented for approximate solutions of inverse problem. Theorems on the stability and coercive stability inequalities for difference schemes are proved. The theoretical statements for the solution of these difference schemes are supported by the results of numerical example in a two-dimensional case. As it can be seen from Tables 13, second order of accuracy difference scheme is more accurate compared with the first order of accuracy difference scheme. Moreover, applying the result of the monograph [29] the high order of accuracy difference schemes for the numerical solution of the boundary value problem (2) can be presented.

Acknowledgment

The authors would like to thank Professor Dr. Allaberen Ashyralyev (Fatih University, Turkey) for his helpful suggestions aimed at the improvement of the present paper.

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