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 Eurasian Math. J., 2012, Volume 3, Number 4, Pages 53–80 (Mi emj105)

Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus

V. G. Kurbatova, I. V. Kurbatovab

a Finance University under the Government of the Russian Federation, Lipetsk, Russia
b Air Force Academy named after professor N. E. Zhukovsky and Y. A. Gagarin, Voronezh, Russia

Abstract: The paper deals with projection methods of approximate solving the problem
$$Fx'=Gx+bu(t),\qquad y=\langle x,d\rangle$$
which consist in passage to the reduced-order problem
$$\widehat F\hat x'=\widehat G\hat x+\hat bu(t),\qquad \hat y=\langle\hat x,\hat d\rangle,$$
where
$$\widehat F=\Lambda FV,\qquad\widehat G=\Lambda GV,\qquad\hat b=\Lambda b,\qquad\hat d=V^*d.$$
It is shown that, if $V$ and $\Lambda$ are constructed on the basis of Krylov's subspaces, a projection method is equivalent to the replacement in the formula expressing the impulse response via the exponential function of the pencil $\lambda\mapsto\lambda F-G$, of the exponential function by its rational interpolation satisfying some interpolation conditions. Special attention is paid to the case when $F$ is not invertible.

Keywords and phrases: Krylov subspaces, Lanczos and Arnoldi methods, differential-algebraic equation, reduced-order system, functional calculus, rational interpolation, operator pencil, pseudoresolvent.

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Bibliographic databases:
MSC: 65L80, 47A58, 41A20
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Citation: V. G. Kurbatov, I. V. Kurbatova, “Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus”, Eurasian Math. J., 3:4 (2012), 53–80

Citation in format AMSBIB
\Bibitem{KurKur12} \by V.~G.~Kurbatov, I.~V.~Kurbatova \paper Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus \jour Eurasian Math. J. \yr 2012 \vol 3 \issue 4 \pages 53--80 \mathnet{http://mi.mathnet.ru/emj105} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3040687} \zmath{https://zbmath.org/?q=an:1267.65098} 

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This publication is cited in the following articles:
1. M. N. Oreshina, “Spectral decomposition of normal operator in real Hilbert space”, Ufa Math. J., 9:4 (2017), 85–96
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