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2015 | OriginalPaper | Buchkapitel

Well-Posedness in Hölder Spaces of Elliptic Differential and Difference Equations

verfasst von : Allaberen Ashyralyev

Erschienen in: Finite Difference Methods,Theory and Applications

Verlag: Springer International Publishing

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Abstract

In the present paper the well-posedness of the elliptic differential equation

$$\begin{aligned} -u^{\prime \prime }(t)+Au(t)=f(t)(-\infty <t<\infty ) \end{aligned}$$

in an arbitrary Banach space E with the general positive operator in Hö lder spaces $C^{\beta }(\mathbb {R},E_{\alpha })$ is established. The exact estimates in Hölder norms for the solution of the problem for elliptic equations are obtained. 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 are studied. The well-posedness of the these difference schemes in the difference analogy of Hölder spaces $C^{\beta }(\mathbb {R}_{\tau }, E_{\alpha })$ are obtained. The almost coercive inequality for solutions in $C(\mathbb {R}_{\tau },E)$ of these difference schemes is established.

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Vorheriges Kapitel A Transparent Boundary Condition for an Elastic Bottom in Underwater Acoustics

Nächstes Kapitel Operator Semigroups for Convergence Analysis

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Titel: Well-Posedness in Hölder Spaces of Elliptic Differential and Difference Equations
verfasst von: Allaberen Ashyralyev
Verlag: Springer International Publishing
Buch: Finite Difference Methods,Theory and Applications
Print ISBN: 978-3-319-20238-9

Electronic ISBN: 978-3-319-20239-6

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-20239-6_3

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