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Zbl 1179.65056
Ashyralyev, Allaberen
High-accuracy stable difference schemes for well-posed NBVP. (English)
[A] 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. Operator Theory: Advances and Applications 191, 229-252 (2009). ISBN 978-3-7643-9920-7/vol.2; ISBN 978-3-7643-9921-4/ebook; ISBN 978-3-7643-9924-5/set

Summary: The single step difference schemes of the high order of accuracy for the approximate solution of the nonlocal boundary value problem (NBVP) $$v'(t) + Av(t) = f(t)(0 \le t \le 1),v\left( 0 \right) = v\left( \lambda \right) + \mu , \quad 0 < \lambda \le 1$$ 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 2000:: *65J08
34G10 Linear ODE in abstract spaces
35K90 Abstract parabolic evolution equations
65M06 Finite difference methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
65L12 Finite difference methods for ODE
47D06 One-parameter semigroups and linear evolution equations
65L20 Stability of numerical methods for ODE

Keywords: parabolic equation; nonlocal boundary value problem; Padé difference schemes; high order of accuracy; well-posedness; coercive inequalities; Banach space; strongly positive operator; abstract parabolic equation; stability

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