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

Allaberen Ashyralyev1 and Ali Sirma2

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.

Abstract

The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy r-modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.