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The long time error analysis in the second order difference type method of an evolutionary integral equation with completely monotonic kernel

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Abstract

We study convergence of the numerical methods in which the second order difference type method is combined with order two convolution quadrature for approximating the integral term of the evolutionary integral equation

$$ u^{\prime}(t)+\int_{0}^{t}\beta(t-s)\,A\,u\,(s)\;ds \, =\,0 ,~~~~ t>0,~~u(0)=u_{0}, $$

which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space H and \(\beta (t)\) is completely monotonic and locally integrable, but not constant. We establish the convergence properties of the discretization in time in the \(l_{t}^{1}(0,\infty ;\,H)\) or \(l_{t}^{\infty }(0,\infty ;\,H)\) norm.

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Authors and Affiliations

  1. Department of Mathematics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Da Xu

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  1. Da Xu
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Correspondence to Da Xu.

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Communicated by: A. Zhou

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Xu, D. The long time error analysis in the second order difference type method of an evolutionary integral equation with completely monotonic kernel. Adv Comput Math 40, 881–922 (2014). https://doi.org/10.1007/s10444-013-9331-2

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