Research Article  Open Access
Allaberen Ashyralyev, Mesut Urun, "Determination of a Control Parameter for the Difference Schrödinger Equation", Abstract and Applied Analysis, vol. 2013, Article ID 548201, 8 pages, 2013. https://doi.org/10.1155/2013/548201
Determination of a Control Parameter for the Difference Schrödinger Equation
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 selfadjoint positive definite operator is constructed. The wellposedness 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 wellposedness 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 selfadjoint positive definite operator was studied. The wellposedness 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 singlestep difference scheme:
The theorem on wellposedness 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 selfadjoint 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 selfadjoint 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 selfadjoint 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 selfadjoint 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 selfadjoint 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 selfadjoint 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 selfadjoint 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 secondorder 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.



5. Conclusion
In the present study, the wellposedness 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 wellposedness 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 singlestep 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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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.