This work considers the variable-exponent fractional diffusion-wave equation, which describes, e.g. the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. Rigorous mathematical and numerical analysis for this model is not available in the literature, partly because the variable-exponent Abel kernel may not be positive definite or monotonic. We overcome these difficulties to design two numerical schemes and derive their stability and error estimate based on the proved solution regularity, with $\alpha(0)$-order and second-order accuracy in time, respectively. Numerical experiments are presented to substantiate the theoretical findings.

翻译：本研究考虑变指数分数阶扩散波方程，该方程描述了例如在具有变化材料特性的粘弹性介质中机械扩散波的传播。由于变指数Abel核可能非正定或非单调，文献中尚未对该模型进行严格的数学与数值分析。我们克服了这些困难，设计了两种数值格式，并基于已证明的解的正则性推导了其稳定性与误差估计，时间精度分别为$\alpha(0)$阶和二阶。数值实验验证了理论结果。