This paper focuses on the distributed-order time-fractional diffusion-wave equations with the Riesz space fractional derivatives. A combined method based on the midpoint quadrature rule, linear B-spline interpolation, and the Galerkin finite element method is proposed to obtain the approximate solution. Two steps are used to calculate the approximate solution to this type of equation. The first step approximates the temporal direction by combining a midpoint quadrature rule and linear B-spline interpolation. In the second step, a Galerkin finite element method in the space direction is applied to compute a full-discrete method. Furthermore, the error estimate has been displayed to demonstrate unconditional stability and convergence. Finally, two numerical examples are reported to show the simplicity and efficiency of the proposed method.

本文重点研究具有 Riesz 空间分数阶导数的分布阶时间分数扩散波方程。提出了一种基于中点求积法则、线性B样条插值和伽辽金有限元法的组合方法来获得近似解。使用两个步骤来计算此类方程的近似解。第一步通过结合中点求积规则和线性 B 样条插值来近似时间方向。第二步，采用空间方向伽辽金有限元法进行全离散法计算。此外，还显示了误差估计以证明无条件稳定性和收敛性。最后，报告了两个数值例子，以表明该方法的简单性和效率。