Approximate solutions of abstract differential equations

Emil Vitásek

Applications of Mathematics (2007)

  • Volume: 52, Issue: 2, page 171-183
  • ISSN: 0862-7940

Abstract

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The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.

How to cite

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Vitásek, Emil. "Approximate solutions of abstract differential equations." Applications of Mathematics 52.2 (2007): 171-183. <http://eudml.org/doc/33282>.

@article{Vitásek2007,
abstract = {The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.},
author = {Vitásek, Emil},
journal = {Applications of Mathematics},
keywords = {abstract differential equations; semigroups of operators; rational approximations; $A$-stability; abstract differential equations; semigroups of operators; rational approximations; -stability},
language = {eng},
number = {2},
pages = {171-183},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximate solutions of abstract differential equations},
url = {http://eudml.org/doc/33282},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Vitásek, Emil
TI - Approximate solutions of abstract differential equations
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 2
SP - 171
EP - 183
AB - The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.
LA - eng
KW - abstract differential equations; semigroups of operators; rational approximations; $A$-stability; abstract differential equations; semigroups of operators; rational approximations; -stability
UR - http://eudml.org/doc/33282
ER -

References

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  1. Fonctions d’une variable réelle (théorie élémentaire), Hermann & Cie, Paris, 1961. (French) (1961) Zbl0131.05001
  2. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987. (1987) Zbl0616.65072MR0878564
  3. Linear Operators, Vol.  I, Interscience, New York-London, 1963. (1963) MR0188745
  4. Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1966. (1966) Zbl0148.12601MR0203473
  5. Overimplicit multistep methods, Apl. Mat. 18 (1973), 399–421. (1973) MR0366041
  6. Functional analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1971. (1971) Zbl0217.16001

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