In this paper, we construct a Crank–Nicolson linear finite difference scheme for a Benjamin–Bona–Mahony equation with a time fractional nonlocal viscous term. The stability and convergence of the proposed numerical scheme are rigorously derived. Theoretical analysis shows that the numerical scheme is convergent in the order of \(O(\tau^{\frac{3}{2}}+h^{2})\), where τ and h are the time and space step sizes. Two numerical experiments are presented to verify that the theoretical analysis is accurate and to demonstrate that the numerical scheme is effective.

In this paper, the transverse vibration of a fractional viscoelastic beam is studied based on the fractional calculus, and the corresponding scheme of a viscoelastic beam is established by using the mixed finite volume element method. The stability and convergence of the algorithm are analyzed. Numerical examples demonstrate the effectiveness of the algorithm. Finally, the values of different parameter sets are tested, and the test results show that both the damping coefficient and the fractional derivative have significant effects on the model. The results of this paper can be used for the damping modeling of viscoelastic structures.

We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both $ |\|·|\| $ and $ L^2 $ norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces $ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $ for $ k = 0 $ and the numerical convergence rates confirm the theoretical results.

This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. We discretize the space derivative via compact difference and approximate the time-integer-order derivative and fraction-order derivative via central difference and L1 interpolation, respectively, to obtain the compact difference formats with fourth-order space precision and $3-\alpha$($1<\alpha <2$)-order time precision. We apply the energy method to analyze the stability and convergence of this difference format. We provide numerical cases, which not only validate the convergence order and feasibility of the given difference format, but also simulate the influence of the damping coefficient on the amplitude of plate vibration.

In this paper, the vibration problem of a beam with a time fractional damping term is studied by the Hermite finite element method, and its fully discrete scheme is obtained. The stability and error estimation of the scheme are analyzed, and it was proved that it is unconditionally stable and has a convergence order of $O(\tau +{\tau }^{3-\alpha }+h^4)$. The validity of the scheme is verified by numerical examples, the effects of fractional derivative order and damping coefficient on beam vibration are analyzed and the superiority of the fractional order model has been demonstrated by comparing with the traditional damping model.