This article is cited in 1 scientific paper (total in 1 paper)
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
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,
\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.
PDF file (586 kB)
MSC: 65L80, 47A58, 41A20
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
\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.
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