求解四阶多项时间分数阶混合扩散-波方程的二阶差分格式

Second-order difference scheme for solving the fourth-order multi-term time fractional mixed diffusion-wave equations

为求解二维四阶多项时间分数阶混合扩散-波方程,基于降阶法将时间分数阶扩散项和分数阶波动项分别转换为时间分数阶积分项和扩散项,并在时间方向分别应用L2-1公式和分片线性插值方法进行离散,对空间四阶导数项也进行降阶处理,建立差分求解格式.利用能量分析法对所得格式的稳定性和收敛性进行严格分析,结果显示其无条件稳定且在时间和空间方向上都是二阶收敛.数值算例证实所得数值格式的精度和有效性.

The finite difference method is considered to solve the two-dimensional fourth-order multi-term time fractional mixed diffusion-wave equations.With the help of the method of order reduction,the multi-term time fractional diffusion-wave terms are converted into the multi-term time fractional integral and diffusion terms respectively.Then,the L2-1formula and the piecewise linear interpolation are applied to discretize the space in the time direction,and the fourth-order derivative term is also reduced in order to establish a difference solution scheme.The stability and convergence of the proposed scheme are rigorously analyzed by the energy method.It is proved that the scheme is unconditionally stable and convergent,with the convergence order of two in both time and space.Finally,the accuracy and effectiveness of the numerical scheme are verified by a numerical example.