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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

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., [124] 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.

tab1
Table 1: Error analysis for the exact solution .
tab2
Table 2: Error analysis for the exact solution .
tab3
Table 3: Error analysis for .

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.

References

  1. M. Dehghan, “Determination of a control parameter in the two-dimensional diffusion equation,” Applied Numerical Mathematics, vol. 37, no. 4, pp. 489–502, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. T. Kimura and T. Suzuki, “A parabolic inverse problem arising in a mathematical model for chromatography,” SIAM Journal on Applied Mathematics, vol. 53, no. 6, pp. 1747–1761, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography,” Inverse Problems, vol. 15, no. 2, pp. 373–397, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Y. S. Eidelman, Boundary value problems for differential equations with parameters [Ph.D. thesis], Voronezh State University, Voronezh, Russia, 1984 (Russian).
  5. A. Hasanov, “Identification of unknown diffusion and convection coefficients in ion transport problems from flux data: an analytical approach,” Journal of Mathematical Chemistry, vol. 48, no. 2, pp. 413–423, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. di Blasio and A. Lorenzi, “Identification problems for parabolic delay differential equations with measurement on the boundary,” Journal of Inverse and Ill-Posed Problems, vol. 15, no. 7, pp. 709–734, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Orlovsky and S. Piskarev, “On approximation of inverse problems for abstract elliptic problems,” Journal of Inverse and Ill-Posed Problems, vol. 17, no. 8, pp. 765–782, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. Orlovsky and S. Piskarev, “The approximation of Bitzadze-Samarsky type inverse problem for elliptic equations with Neumann conditions,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 118–131, 2013.
  9. A. Ashyralyev, “On the problem of determining the parameter of a parabolic equation,” Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1397–1408, 2011. View at MathSciNet
  10. V. Serov and L. Päivärinta, “Inverse scattering problem for two-dimensional Schrödinger operator,” Journal of Inverse and Ill-Posed Problems, vol. 14, no. 3, pp. 295–305, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. Ashyralyev and B. Hicdurmaz, “On the numerical solution of fractional Schrödinger differential equations with the Dirichlet condition,” International Journal of Computer Mathematics, vol. 89, no. 13-14, pp. 1927–1936, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. A. Ashyralyev, A. S. Erdogan, and O. Demirdag, “On the determination of the right-hand side in a parabolic equation,” Applied Numerical Mathematics, vol. 62, no. 11, pp. 1672–1683, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. S. Erdogan and H. Uygun, “A note on the inverse problem for a fractional parabolic equation,” Abstract and Applied Analysis, vol. 2012, Article ID 276080, 26 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Ch. Ashyralyyev and M. Dedeturk, “A finite difference method for the inverse elliptic problem with the Dirichlet condition,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 132–155, 2013.
  16. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Avdonin, S. Lenhart, and V. Protopopescu, “Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method,” Inverse Problems, vol. 18, no. 2, pp. 349–361, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. Eskin and J. Ralston, “Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy,” Communications in Mathematical Physics, vol. 173, no. 1, pp. 199–224, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Mercado, A. Osses, and L. Rosier, “Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights,” Inverse Problems, vol. 24, no. 1, Article ID 015017, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. T. Nadareishvili and A. Khelashvili, “Pragmatic SAE procedure in the Schrodinger equation for the inverse-square-like potentials,” High Energy Physics—Theory, vol. 93, pp. 1–26, 2012.
  21. H. Nakatsuji, “Inverse Schrödinger equation and the exact wave function,” Physical Review A, vol. 65, no. 5, pp. 1–15, 2002.
  22. A. Ashyralyev and A. Sirma, “Nonlocal boundary value problems for the Schrödinger equation,” Computers & Mathematics with Applications, vol. 55, no. 3, pp. 392–407, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. Ashyralyev and A. Sirma, “Modified Crank-Nicolson difference schemes for nonlocal boundary value problem for the Schrödinger equation,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 584718, 15 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. A. Ashyralyev and A. Sirma, “A note on the numerical solution of the semilinear Schrödinger equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2507–e2516, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. Ashyralyev and M. Urun, “Determination of a control parameter for the Schrödinger equation,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 156–166, 2013.
  26. P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, Russia, 1975 (Russian).