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High-accuracy stable difference schemes for well-posed NBVP. (English) Zbl 1179.65056
Adamyan, Vadim (ed.) et al., Modern analysis and applications. The Mark Krein centenary conference. Volume 2: Differential operators and mechanics. Papers based on invited talks at the international conference on modern analysis and applications, Odessa, Ukraine, April 9–14, 2007. Basel: Birkhäuser (ISBN 978-3-7643-9920-7/v. 2; 978-3-7643-9921-4/ebook; 978-3-7643-9924-5/set). Operator Theory: Advances and Applications 191, 229-252 (2009).

Summary: The single step difference schemes of the high order of accuracy for the approximate solution of the nonlocal boundary value problem (NBVP)

${v}^{\text{'}}\left(t\right)+Av\left(t\right)=f\left(t\right)\left(0\le t\le 1\right),v\left(0\right)=v\left(\lambda \right)+\mu ,\phantom{\rule{1.em}{0ex}}0<\lambda \le 1$

or the differential equation in an arbitrary Banach space $E$ with the strongly positive operator $A$ are presented. The construction of these difference schemes is based on the Padé difference schemes for the solutions of the initial-value problem for the abstract parabolic equation and the high order approximation formula for

$v\left(0\right)=v\left(\lambda \right)+\mu ·$

The stability, the almost coercive stability and coercive stability of these difference schemes are established.

##### MSC:
 65J08 Abstract evolution equations (numerical methods) 34G10 Linear ODE in abstract spaces 35K90 Abstract parabolic equations 65M06 Finite difference methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 65L12 Finite difference methods for ODE (numerical methods) 47D06 One-parameter semigroups and linear evolution equations 65L20 Stability and convergence of numerical methods for ODE