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Zbl 0801.65135
Thomée, Vidar; Wahlbin, Lars B.
Long-time numerical solution of a parabolic equation with memory.
(English)
[J] Math. Comput. 62, No.206, 477-496 (1994). ISSN 0025-5718; ISSN 1088-6842/e

The stability of temporal discretization is investigated for integral equations of the form $\partial\sb t u + Au = \int\sp t\sb 0 b(t - s) Bu(s) ds + f(t)$, where $\partial\sb t + A$ is a parabolic operator on a Hilbert space. The principal question is whether or not solutions remain bounded as $t \to \infty$. The method of analysis is by energy inequalities, but for the special case $B = A$ more detailed results are obtained from eigenfunction expansions. The question of quadrature schemes for the integral operator is also discussed.
[G.Hedstrom (Livermore)]
MSC 2000:
*65R20 Integral equations (numerical methods)
45K05 Integro-partial differential equations
45N05 Integral equations in abstract spaces

Keywords: long-time numerical solution; parabolic equation with memory; stability; Hilbert space; energy inequalities; eigenfunction expansions; quadrature schemes

Cited in: Zbl 0958.45004

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