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

Highlights：

• (2) 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 is evaluated. 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.

• (3) Numerical experiments confirm the theoretic findings and show that the computing efficiency is nearly 103 times lower than the commonly used central difference does.

摘要

•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.•(2) 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 is evaluated. 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.•(3) Numerical experiments confirm the theoretic findings and show that the computing efficiency is nearly 103 times lower than the commonly used central difference does.

论文关键词：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