本文针对一类四阶非线性时间分数阶双曲方程构造了一种具有时空二阶精度的数值算法。首先引入一个低阶参数β=α-1将一个α ∈ (1, 2) 的Caputo分数阶导数降为β∈ (0, 1) 的Caputo分数阶导数, 其次引入两个辅助函数变量v=ut , ζ=Δu将四阶非线性时间分数阶双曲方程转化为一个包含积分项的低阶非线性耦合系统。在时间方向上利用L2-1σ公式对其分数阶导数项进行离散, 同时利用复化梯形公式离散积分项, 在空间上采用传统Galerkin有限元方法, 进一步得到该低价非线性耦合系统的全离散有限元格式。以二维区域上的均匀矩形剖分为例, 利用双线性基函数给出了刚度矩阵和荷载矩阵的具体形式, 得到了全离散数值格式的详细算法实现过程。最后分别对一维和二维算例进行数值模拟, 对比于原方程该低阶系统能够同时观察三个未知量u, v, ζ的变化, 并且详细地列出不同参数和不同时空步长下这三个未知量的误差、图像和收敛阶, 充分地验证了该算法的可行性和有效性。

In this paper, a numerical algorithm with space-time second-order accuracy is constructed for a fourth-order nonlinear time fractional hyperbolic equation. Firstly, a low order parameter β =α-1 is introduced to reduce the Caputo fractional derivative ofα ∈ 1, 2 to the Caputo fractional derivative of β ∈ 0, 1 . Secondly, two auxiliary variables v=ut , ζ=Δu are introduced to transform. the fourth-order nonlinear time fractional hyperbolic equation into a low-order nonlinear coupled system with integral term. The fractional derivative term is discretized in the time direction by L2-1σ formula, and the integral term is discretized by the complex trapezoidal formula. Using the traditional Galerkin finite element method in space, the fully discrete finite element scheme of the low-order nonlinear coupling system is obtained. Taking the uniform. rectangular division on a two-dimensional region as an example, the specific forms of stiffness matrix and load matrix are given by the bilinear basis function, and the detailed algorithm implementation process of the fully discrete numerical scheme is obtained. Finally, numerical simulations are carried out for one-dimensional and two-dimensional examples. Compared with the original equation, the low-order system can observe the changes of three unknown variables u, v, ζ at the same time, and the errors, images and convergence orders of the three unknown variables are listed in detail under different parameters and different space-time steps, which fully verifies the feasibility and effectiveness of the algorithm.