We analyze a variable-order time-fractional wave equation, which models, e.g., the vibration of a membrane in a viscoelastic environment. We prove that the solutions to the variable-order ordinary differential equations in the spectral decomposition of the solution to the fractional wave equation exhibit power-law decaying characteristics and overcome the difficulty that its solution operator does not have an exponential decay in contrast to its variable-order fractional diffusion analogue.

We prove an optimal-order error estimate of a numerical discretization of the variable-order fractional wave equation only under regularity assumptions of the data of the model but with no smoothness assumption of its solution. As the solution exhibits initial weak singularity, the local truncation error is suboptimal. A conventional analysis gives a suboptimal-order estimate. We develop a new technique to derive the desired optimal-order convergence rate. We also conduct numerical experiments to substantiate the mathematically proved findings.

我们分析了一个可变阶时间分数波动方程，该方程模拟了例如膜在粘弹性环境中的振动。我们证明了分数波动方程解的谱分解中变阶常微分方程的解表现出幂律衰减特性，克服了其解算子与其变量相比不具有指数衰减的困难。 -阶分数扩散模拟。

我们仅在模型数据的规律性假设下证明了变阶分数波动方程的数值离散化的最优阶误差估计，但没有对其解的平滑假设。由于解表现出初始弱奇异性，局部截断误差是次优的。常规分析给出次优阶估计。我们开发了一种新技术来推导出所需的最优阶收敛速度。我们还进行了数值实验来证实数学证明的发现。