Abstract and Applied Analysis
Volume 2010 (2010), Article ID 705172, 17 pages
doi:10.1155/2010/705172
Research Article

On Second Order of Accuracy Difference Scheme of the Approximate Solution of Nonlocal Elliptic-Parabolic Problems

Allaberen Ashyralyev1,2 and Okan Gercek3,4

1Department of Mathematics, Fatih University, Buyukcekmece, Istanbul 34500, Turkey
2Department of Mathematics, ITTU, Ashgabat 744012, Turkmenistan
3Vocational School, Fatih University, Buyukcekmece, Istanbul 34500, Turkey
4Department of Mathematics, Yildiz Technical University, 34210 Istanbul, Turkey

Received 16 February 2010; Accepted 15 May 2010

Academic Editor: Ağacik Zafer

Copyright © 2010 Allaberen Ashyralyev and Okan Gercek. 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

A second order of accuracy difference scheme for the approximate solution of the abstract nonlocal boundary value problem −d2u(t)/dt2+Au(t)=g(t), (0≤t≤1), du(t)/dt−Au(t)=f(t), (−1≤t≤0), u(1)=u(−1)+μ for differential equations in a Hilbert space H with a self-adjoint positive definite operator A is considered. The well posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained and a numerical example is presented.