Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 584718, 15 pages
doi:10.1155/2009/584718
Research Article

Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation

1Department of Mathematics, Fatih University, 34500 Büyükcekmece, Istanbul, Turkey
2Department of Mathematics and Computer Sciences, Bahcesehir University, Besiktas, 34353 Istanbul, Turkey

Received 26 November 2008; Revised 30 March 2009; Accepted 19 June 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Allaberen Ashyralyev and Ali Sirma. 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. E. Mayfield, Non-reflective boundary conditions for Schrödinger's equation, Ph.D. thesis, University of Rhode Island, Kingston, RI, USA, 1989.
  2. D. G. Gordeziani and G. A. Avalishvili, “Time-nonlocal problems for Schrödinger-type equations. I. Problems in abstract spaces,” Differential Equations, vol. 41, no. 5, pp. 670–677, 2005. View at Zentralblatt MATH · View at MathSciNet
  3. D. G. Gordeziani and G. A. Avalishvili, “Time-nonlocal problems for Schrödinger-type equations. II. Results for specific problems,” Differential Equations, vol. 41, no. 6, pp. 813–819, 2005. View at Zentralblatt MATH · View at MathSciNet
  4. H. Han, J. Jin, and X. Wu, “A finite-difference method for the one-dimensional time-dependent Schrödinger equation on unbounded domain,” Computers & Mathematics with Applications, vol. 50, no. 8-9, pp. 1345–1362, 2005. View at Zentralblatt MATH · View at MathSciNet
  5. J. Bourgain, “Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential,” Communications in Mathematical Physics, vol. 204, no. 1, pp. 207–247, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. X. Antoine, C. Besse, and V. Mouysset, “Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions,” Mathematics of Computation, vol. 73, no. 248, pp. 1779–1799, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Ashyralyev, S. Piskarev, and L. Weis, “On well-posedness of difference schemes for abstract parabolic equations in Lp([0,1],E) spaces,” Numerical Functional Analysis and Optimization, vol. 23, no. 7-8, pp. 669–693, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. 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 Zentralblatt MATH · View at MathSciNet
  9. A. Ashyralyev and A. Sirma, “A note on the modified Crank-Nicolson difference schemes for Schrödinger equation,” in Complex Analysis and Potential Theory (Proceedings of the Conference Satellite to ICM 2006), pp. 256–271, World Scientific Press, River Edge, NJ, USA, 2007. View at Zentralblatt MATH
  10. A. Ashyralyev, “Well-posedness of the modified Crank-Nicholson difference schemes in Bochner spaces,” Discrete and Continuous Dynamical Systems. Series B, vol. 7, no. 1, pp. 29–51, 2007. View at MathSciNet
  11. P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izland. Voronezh. Gosud. Univ., Voronezh, Russia, 1975.
  12. A. Ashyralyev, A. S. Erdogan, and N. Arslan, “On the modified Crank-Nicholson difference schemes for parabolic equation with non-smooth data arising in biomechanics,” Communications in Numerical Methods in Engineering. View at Publisher · View at Google Scholar