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Long-time numerical solution of a parabolic equation with memory. (English) Zbl 0801.65135

The stability of temporal discretization is investigated for integral equations of the form \(\partial_ t u + Au = \int^ t_ 0 b(t - s) Bu(s) ds + f(t)\), where \(\partial_ t + A\) 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.

MSC:

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
45N05 Abstract integral equations, integral equations in abstract spaces
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