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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)
$v'(t) + Av(t) = f(t)(0 \leq t \leq 1),v\left( 0 \right) = v\left( \lambda \right) + \mu , \quad 0 < \lambda \leq 1$
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.
For the entire collection see [Zbl 1169.47002].

##### MSC:
 65J08 Numerical solutions to abstract evolution equations 34G10 Linear differential equations in abstract spaces 35K90 Abstract parabolic equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65L12 Finite difference and finite volume methods for ordinary differential equations 47D06 One-parameter semigroups and linear evolution equations 65L20 Stability and convergence of numerical methods for ordinary differential equations
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