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

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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 ≤ j ≤ n, 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.

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Funding

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

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Authors and Affiliations

  1. Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Department of Mathematics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Da Xu

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Correspondence to Da Xu.

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Xu, D. Second-order difference approximations for Volterra equations with the completely monotonic kernels. Numer Algor 81, 1003–1041 (2019). https://doi.org/10.1007/s11075-018-0580-5

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  • DOI: https://doi.org/10.1007/s11075-018-0580-5

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