Numerical analysis for compact difference scheme of fractional viscoelastic beam vibration models

Qing Li and Huanzhen Chen

Applied Mathematics and Computation, 2022, vol. 427, issue C

Abstract: In this article, a compact difference method is proposed for fractional viscoelastic beam vibration in stress-displacement form. The solvability, the unconditional stability and the convergence rates of second-order in time and fourth-order in space are rigorously proved for the fractional stress v and the displacement u, respectively, under a mild assumption on the loading f. Numerical experiments are given to verify the theoretic findings. One of the main contributions of this article is to evaluate the positive lower- and upper-bound of the eigenvalues of the Toeplitz matrix Λ generated from the weighted Grünwald difference operator for fractional integral operators, and thus prove that the matrix Λ is positive definite and can induce a norm in a vector space. This finding improves significantly the existing semi-positive definiteness theory of the matrix Λ for fractional differential operators and facilitates the proof of the stability and convergence for the stress v.

Keywords: Fractional viscoelastic beam vibration; The weighted Grünwald difference operator; Fourth-order compact finite difference scheme; Lower and upper bounds of eigenvalues; Stability and convergence analysis (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:427:y:2022:i:c:s0096300322002259

DOI: 10.1016/j.amc.2022.127146

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