A finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Consistency, stability and convergence of the method are established. Numerical experiments illustrating the effectiveness of the theoretical analysis are provided.
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.
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.
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−α(1<α<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.
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