This work presents two numerical schemes for the variable-exponent fractional diffusion-wave equation, which describes, e.g. the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. The main difficulty we overcome lies in that the variable-exponent Abel kernel may not be positive definite or monotonic, and the stability and error estimate of both schemes are proved, with $\alpha(0)$-order and second-order accuracy in time, respectively. Numerical experiments are presented to substantiate the theoretical findings.

翻译：本文针对变指数分数阶扩散波动方程提出了两种数值格式，该方程可用于描述例如变材料属性粘弹性介质中机械扩散波的传播过程。我们克服的主要困难在于变指数阿贝尔核可能不具备正定性或单调性，并证明了两种格式的稳定性与误差估计，其时间精度分别达到$\alpha(0)$阶和二阶精度。数值实验验证了理论分析结果。