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ЖУРНАЛЫ // Eurasian Mathematical Journal // Архив

Eurasian Math. J., 2012, том 3, номер 4, страницы 53–80 (Mi emj105)

Эта публикация цитируется в 3 статьях

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

Аннотация: 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.

Ключевые слова и фразы: Krylov subspaces, Lanczos and Arnoldi methods, differential-algebraic equation, reduced-order system, functional calculus, rational interpolation, operator pencil, pseudoresolvent.

MSC: 65L80, 47A58, 41A20

Поступила в редакцию: 20.11.2012

Язык публикации: английский



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