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

Ashyralyev, Allaberen; Sobolevskiĭ, Pavel E.

doi:https://doi.org/10.1155/DDNS/2006/75153

Discrete Dynamics in Nature and Society

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Volume 2006 | Article ID 075153 | https://doi.org/10.1155/DDNS/2006/75153

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

Allaberen Ashyralyev¹and Pavel E. Sobolevskiĭ^2,3

Received22 May 2005

Accepted14 Aug 2005

Published07 Mar 2006

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.

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Copyright

Copyright © 2006 Allaberen Ashyralyev and Pavel E. Sobolevskiĭ. 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.

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