Approximation of abstract linear integrodifferential equations

Hirokazu Oka; Naoki Tanaka

Studia Mathematica (2000)

  • Volume: 140, Issue: 1, page 1-14
  • ISSN: 0039-3223

Abstract

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This paper is devoted to the approximation of abstract linear integrodifferential equations by finite difference equations. The result obtained here is applied to the problem of convergence of the backward Euler type discrete scheme.

How to cite

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Oka, Hirokazu, and Tanaka, Naoki. "Approximation of abstract linear integrodifferential equations." Studia Mathematica 140.1 (2000): 1-14. <http://eudml.org/doc/216754>.

@article{Oka2000,
abstract = {This paper is devoted to the approximation of abstract linear integrodifferential equations by finite difference equations. The result obtained here is applied to the problem of convergence of the backward Euler type discrete scheme.},
author = {Oka, Hirokazu, Tanaka, Naoki},
journal = {Studia Mathematica},
keywords = {approximation of integrodifferential equation; approximation of; Banach space; resolvent operator; semigroup of class $(C_0)$; linear operators; Cauchy problem; finite differences; abstract linear integrodifferential equations},
language = {eng},
number = {1},
pages = {1-14},
title = {Approximation of abstract linear integrodifferential equations},
url = {http://eudml.org/doc/216754},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Oka, Hirokazu
AU - Tanaka, Naoki
TI - Approximation of abstract linear integrodifferential equations
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 1
SP - 1
EP - 14
AB - This paper is devoted to the approximation of abstract linear integrodifferential equations by finite difference equations. The result obtained here is applied to the problem of convergence of the backward Euler type discrete scheme.
LA - eng
KW - approximation of integrodifferential equation; approximation of; Banach space; resolvent operator; semigroup of class $(C_0)$; linear operators; Cauchy problem; finite differences; abstract linear integrodifferential equations
UR - http://eudml.org/doc/216754
ER -

References

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  1. [1] G. Chen and R. Grimmer, Integral equations as evolution equations, J. Differential Equations 45 (1982), 53-74. Zbl0449.45008
  2. [2] R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces, Comput. Math. Appl. 11 (1985), 189-205. Zbl0569.45020
  3. [3] H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), 160-180. Zbl0689.47014
  4. [4] J. Kisyński, A proof of the Trotter-Kato theorem on approximation of semigroups, Colloq. Math. 18 (1967), 181-184. Zbl0152.33905
  5. [5] T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Funct. Anal. 3 (1969), 354-375. Zbl0174.18401
  6. [6] H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 (1995), 193-232. Zbl0846.45005
  7. [7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. 
  8. [8] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. 
  9. [9] N. Tanaka, Approximation of integrated semigroups by ``integrated'' discrete parameter semigroups, Semigroup Forum 55 (1997), 57-67. Zbl0897.47038
  10. [10] V. Thomée and L. B. Wahlbin, Long-time numerical solution of a parabolic equation with memory, Math. Comp. 62 (1994), 477-496. Zbl0801.65135
  11. [11] H. F. Trotter, Approximation of semigroups of operators, Pacific J. Math. 8 (1958), 887-919. Zbl0099.10302

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