Discrete Dynamics in Nature and Society
Volume 2005 (2005), Issue 2, Pages 183-213
doi:10.1155/DDNS.2005.183

Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations

Allaberen Ashyralyev1,2 and Pavel E. Sobolevskii3,4

1Department of Mathematics, Fatih University, Buyukcekmece, Istanbul 39400, Turkey
2International Turkmen-Turkish University, Ashgabat 744012, Turkmenistan
3Institute of Mathematics, Federal University of Ceará, Fortaleza 60020-181, Ceará, Brazil
4Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel

Received 30 March 2004

Copyright © 2005 Hindawi Publishing Corporation. 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

We consider the abstract Cauchy problem for differential equation of the hyperbolic type v″(t)+Av(t)=f(t) (0≤t≤T), v(0)=v0, v′(0)=v′0 in an arbitrary Hilbert space H with the selfadjoint positive definite operator A. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of this problem are presented. The stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.