Thomée, Vidar; Wahlbin, Lars B. Long-time numerical solution of a parabolic equation with memory. (English) Zbl 0801.65135 Math. Comput. 62, No. 206, 477-496 (1994). 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. Reviewer: G.Hedstrom (Livermore) Cited in 1 ReviewCited in 24 Documents MSC: 65R20 Numerical methods for integral equations 45K05 Integro-partial differential equations 45N05 Abstract integral equations, integral equations in abstract spaces Keywords:long-time numerical solution; parabolic equation with memory; stability; Hilbert space; energy inequalities; eigenfunction expansions; quadrature schemes PDFBibTeX XMLCite \textit{V. Thomée} and \textit{L. B. Wahlbin}, Math. Comput. 62, No. 206, 477--496 (1994; Zbl 0801.65135) Full Text: DOI