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Numerical Algorithms

, Volume 81, Issue 3, pp 1003–1041 | Cite as

Second-order difference approximations for Volterra equations with the completely monotonic kernels

Original Paper
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Abstract

The second-order difference type methods are studied for the solution of the problem
$$u^{\prime}(t)+{{\int}_{0}^{t}} (t-\tau)u (\tau)d\tau = 0 , t>0, u(0) = u_{0}, $$
with \( L(t)=\sum \limits _{j = 1}^{n} a_{j}(t) L_{j} \). The operators Lj are densely defined positive self-adjoint linear operator on a Hilbert space H and have spectral decompositions with respect to a common resolution of the identity {Eλ} in H. Here, the kernel functions aj(t), 1 ≤ jn, are completely monotonic on (0, ∞) and locally integrable, but not constant. The convergence properties of the time discretization are proven in the weighted \( l^{1}(\rho ;0,\infty ; \mathbf {H} ) \) and \( l^{\infty }(\rho ; 0, \infty ; \mathbf {H} ) \) norm, where ρ is a given weighted function.

Keywords

Integro-differential equations Completely monotonic kernels The second-order backward difference time-stepping schemes Weighted l1 asymptotic convergence behavior 

Mathematics Subject Classification (2010)

45K05 65D30 34K30 65L07 65M12 

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Notes

Funding information

This work was supported in part by the National Natural Science Foundation of China, contract grant number 11671131.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China

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