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

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.