Discrete Dynamics in Nature and Society 
Volume 2006 (2006), Article ID 75153, 12 pages
doi:10.1155/DDNS/2006/75153

Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

Allaberen Ashyralyev1 and Pavel E. Sobolevskiĭ2,3

 1Department of Mathematics, Fatih University, Istanbul, Turkey
2Institute of Mathematics, Universidade Federal do Ceara, Brazil
3Institute of Mathematics, Hebrew University, Jerusalem, Israel
Received 22 May 2005; Accepted 14 August 2005

Abstract

It is well known the differential equation −u″(t)+Au(t)=f(t)(−∞<t<∞) in a general Banach space E with the positive operator A is ill-posed in the Banach space C(E)=C((−∞,∞),E) of the bounded continuous functions ϕ(t) defined on the whole real line with norm ‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions in C(τ,E) of these difference schemes is established.