Discrete Dynamics in Nature and Society 
Volume 2007 (2007), Article ID 57491, 25 pages
doi:10.1155/2007/57491
Research Article

Stability of a Second Order of Accuracy Difference Scheme for Hyperbolic Equation in a Hilbert Space

Allaberen Ashyralyev1 and Mehmet Emir Koksal2,3

 1Department of Mathematics, Fatih University, Buyukcekmece 34900, Istanbul, Turkey
2Graduate Institute of Sciences and Engineering, Fatih University, Buyukcekmece 34900, Istanbul, Turkey
3Department of Mathematics, Gebze Institute of Technology, 41400, Gebze/Kocaeli, Turkey
Received 7 June 2007; Accepted 16 September 2007

Abstract

The initial-value problem for hyperbolic equation d2u(t)/dt2+A(t)u(t)=f(t)(0≤t≤T), u(0)=ϕ,u′(0)=ψ in a Hilbert space H with the self-adjoint positive definite operators A(t) is considered. The second order of accuracy difference scheme for the approximately solving this initial-value problem is presented. The stability estimates for the solution of this difference scheme are established.