Abstract and Applied Analysis

Volume 2013 (2013), Article ID 548201, 8 pages

http://dx.doi.org/10.1155/2013/548201

Research Article

Determination of a Control Parameter for the Difference Schrödinger Equation

Allaberen Ashyralyev1,2 and Mesut Urun1,3

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

²ITTU, 32 Gerogly Street, 74400 Ashgabat, Turkmenistan

³Department of Mathematics, Murat Education Institution, 34353 Istanbul, Turkey

Received 28 July 2013; Accepted 18 September 2013

Academic Editor: Abdullah Said Erdogan

Copyright © 2013 Allaberen Ashyralyev and Mesut Urun. 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.

The first order of accuracy difference scheme for the numerical solution of the boundary value problem for the differential equation with parameter , , , , , in a Hilbert space with self-adjoint positive definite operator is constructed. The well-posedness of this difference scheme is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained.

1. Introduction: Difference Scheme

The theory and applications of well-posedness of inverse problems for partial differential equations have been studied extensively in a large cycle of papers (see, e.g., [1–24] and the references therein).

Our goal in this paper is to investigate Schrödinger equations with parameter. In the paper [25], the boundary value problem for the differential equation with parameter

in a Hilbert space with self-adjoint positive definite operator was studied. The well-posedness of this problem was established. The stability inequalities for the solution of three determinations of control parameter problems for the Schrödinger equation were obtained. In the present paper, the first order of accuracy Rothe difference scheme

for the approximate solution of the boundary value problem (1) for the differential equation with parameter is presented. It is easy to see that

where is the solution of the following single-step difference scheme:

The theorem on well-posedness of difference problem (2) is proved. In practice, the stability inequalities for the solution of difference schemes for the approximate solution of three different types of control parameter problems are obtained.

The paper is organized as follows. Section 1 is the introduction. In Section 2, the main theorem on stability of difference problem (2) is established. In Section 3, theorems on the stability inequalities for the solution of difference schemes for the approximate solution of three different types of control parameter problems are obtained. In Section 4, numerical results are given. Finally, Section 5 is the conclusion.

2. The Main Theorem on Stability

In this section, we will study the stability of difference scheme (2).

Let be the uniform grid space with step size , where is a fixed positive integer. Throughout the present paper, denotes the linear space of grid functions with values in the Hilbert space . Let be the Banach space of bounded grid functions with the norm

Let us start with a lemma we need below. We denote that is the step operator of problem (2).

Lemma 1. Assume that is a positive definite self-adjoint operator. The operator has an inverse and the following estimate is satisfied:

Proof. The proof of estimate (6) is based on the triangle inequality and the estimate

Now, let us obtain the formula for the solution of problem (2). It is clear that the first order of accuracy difference scheme

has a solution and the following formula

is satisfied. Applying formula (9) and the boundary condition

we can write

Since

we have that

By Lemma 1, we get

Using (9) and (14), we get

Since

we have that

Hence, difference scheme (2) is uniquely solvable and for the solution, formulas (14) and (17) hold.

Theorem 2. Suppose that the assumption of Lemma 1 holds. Let . Then, for the solution of difference scheme (2) in , the estimates

hold, where is independent of , , , and .

Proof. From formulas (9) and (14), it follows that

Using this formula, the triangle inequality, and estimate (6), we obtain

Estimate (18) is proved. Using formula (17), the triangle inequality, and estimate (6), we obtain

for any . From that, it follows estimate (19). This completes the proof of Theorem 2.

3. Applications

Now, we consider the simple applications of main Theorem 2.

First, the boundary value problem for the Schrödinger equation

is considered. Problem (23) has a unique smooth solution for the smooth functions , , , , , (), and (, ). This allows us to reduce the boundary value problem (23) to the boundary value problem (1) in a Hilbert space with a self-adjoint positive definite operator defined by formula

with domain

The discretization of problem (23) is carried out in two steps. In the first step, we define the grid space

Let us introduce the Hilbert space of the grid functions

defined on , equipped with the norm

To the differential operator defined by formula (24), we assign the difference operator by the formula

acting in the space of grid functions satisfying the conditions , . It is well known that is a self-adjoint positive definite operator in . With the help of , we reach the boundary value problem

In the second step, we replace (30) with the difference scheme (2)

Theorem 3. The solution pairs of problem (31) satisfy the stability estimates

where and do not depend on , , and , . Here, is the grid space of grid functions defined on with norm

The proof of Theorem 3 is based on formulas for and and the symmetry property of operator .

Second, let be the unit open cube in the -dimensional Euclidean space with boundary , . In , the boundary value problem for the multidimensional Schrödinger equation

is considered. Here, (), (, ), and , () are given smooth functions.

We consider the Hilbert space of all square integrable functions defined on , equipped with the norm

Problem (34) has a unique smooth solution , for the smooth functions , , , and . This allows us to reduce the problem (34) to the boundary value problem (1) in the Hilbert space with a self-adjoint positive definite operator defined by the formula

with domain

The discretization of problem (34) is carried out in two steps. In the first step, we define the grid space

and introduce the Hilbert space of the grid functions

defined on , equipped with the norm

To the differential operator defined by formula (36), we assign the difference operator by the formula

where is known as self-adjoint positive definite operator in , acting in the space of grid functions satisfying the conditions for all . With the help of the difference operator , we arrive to the following boundary value problem:

for an infinite system of ordinary differential equations.

The first order of accuracy difference scheme for the solution of problem (42) is

Theorem 4. The solution pairs of problem (43) satisfy the stability estimates

where and do not depend on , , and , . Here, is the grid space of grid functions defined on with norm

The proof of Theorem 4 is based on Theorem 3 and the symmetry property of the operator is defined by formula (34) and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .

Theorem 5. For the solutions of the elliptic difference problem [26]

the following coercivity inequality holds:

where does not depend on and .

Third, in , the boundary value problem for the multidimensional Schrödinger equation

with the Neumann condition is considered. Here, is the normal vector to , , and , (, ), and , () are given smooth functions.

Problem (48) has a unique smooth solution for the smooth functions , , , and . This allows us to reduce the problem (48) to the boundary value problem (1) in the Hilbert space with a self-adjoint positive definite operator defined by formula

with domain

The discretization of problem (48) is carried out in two steps. In the first step, we define the difference operator by the formula

where is known as self-adjoint positive definite operator in , acting in the space of grid functions satisfying the conditions for all . Here, is the approximation of the operator ·. With the help of the difference operator , we arrive to the following boundary value problem:

for an infinite system of ordinary differential equations.

The first order of accuracy difference scheme for the solution of problem (52) is

Theorem 6. The solution pairs of problem (53) satisfy the stability estimates

where and do not depend on , , and .

The proof of Theorem 6 is based on Theorem 2 and the symmetry property of the operator is defined by formula (51) and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .

Theorem 7. For the solution of the elliptic difference problem [26]

the following coercivity inequality holds:

where does not depend on and .

4. Numerical Results

In present section, for numerical analysis, the following boundary value problem

is considered. The exact solution of problem (57) is and .

The first order of accuracy difference scheme

for the numerical solution of problem (57) is constructed.

For obtaining the values of at the grid points, we will use the following equation:

where , , and is the solution of the first order of accuracy difference scheme

generated by difference scheme (58).

Using the difference scheme (60), we obtain system of linear equations and we can write them in the matrix form as

where

Here,

So, we have the second-order difference equation with respect to with matrix coefficients. Using the modified Gauss elimination method, we can obtain , , .

For the solution of the matrix equations, we seek the solution of the form

where and , , are calculated as

where is and is zero matrix.

Then, using (59), values of at grid points are obtained. Replacing in (58), we get system of linear equations and it can be written in the matrix form

where

Here,

Using the modified Gauss elimination method again, we can obtain , , .

We will give the results of the numerical analysis. The numerical solutions are recorded for different values of and and represents the numerical solutions of the difference scheme at . Table 1 is constructed for , , and , respectively and the errors are computed by the following formula:

For their comparison, Table 2 is constructed when errors are computed by

Table 3 is constructed for the error of at the nodes in maximum norm.

Table 1: Error analysis for the exact solution .

Method

1st order of accuracy d.s.
0.0024
0.0012
6.0463 × 10⁻⁴

Table 2: Error analysis for the exact solution .

Method

1st order of accuracy d.s.
0.0019
9.5692 × 10⁻⁴
4.8241 × 10⁻⁴

Table 3: Error analysis for .

Method

1st order of accuracy d.s.
0.0145
0.0072
0.0036

5. Conclusion

In the present study, the well-posedness of difference problem for the approximate solution of determination of a control parameter for the Schrödinger equation is established. In practice, the stability inequalities for the solution of difference schemes of the approximate solution of three different types of control parameter problems are obtained. The well-posedness of the boundary value problem (1) is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained. Moreover, applying the result of the monograph [15], the high order of accuracy single-step difference schemes for the numerical solution of the boundary value problem (1) can be presented. Of course, the stability inequalities for the solution of these difference schemes have been established without any assumptions about the grid steps.

Conflict of Interests

The authors declare that they have no conflict of interests.

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