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

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  • Li, Qing
  • Chen, Huanzhen

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.

Suggested Citation

  • Li, Qing & Chen, Huanzhen, 2022. "Numerical analysis for compact difference scheme of fractional viscoelastic beam vibration models," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    • Handle: RePEc:eee:apmaco:v:427:y:2022:i:c:s0096300322002259
    DOI: 10.1016/j.amc.2022.127146
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    References listed on IDEAS

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    1. Yu, Chunxiao & Zhang, Jie & Chen, Yiming & Feng, Yujing & Yang, Aimin, 2019. "A numerical method for solving fractional-order viscoelastic Euler–Bernoulli beams," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 275-279.
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    Cited by:

    1. Li, Yiqun & Wang, Hong, 2023. "A finite element approximation to a viscoelastic Euler–Bernoulli beam with internal damping," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 138-158.

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