About this Journal Submit a Manuscript Table of Contents
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.

Linked 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).