We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency-dependent power law. First we prove the existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time-stepping scheme based on the finite element method in space and a combination of convolution quadrature and second-order central differences in time. We conduct a full error analysis of the mixed time discretization and in turn the fully space-time discretized scheme. Error estimates are given for both smooth solutions and solutions with a singularity at $t = 0$ of a type that is typical for equations involving fractional time derivatives. A number of numerical results are presented to support the error analysis.

中文翻译：

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服从频率幂律的有损介质波动方程的数值分析

我们研究了一个具有非局部时间分数阻尼项的波动方程，该方程模拟了以频率相关的幂律为特征的声衰减效应。首先，我们证明了这个方程的唯一解的存在，特别注意了分数导数的处理。然后，我们基于空间中的有限元方法以及卷积正交和二阶时间中心差的组合，推导出了一个显式时间步长方案。我们对混合时间离散化以及完全时空离散化方案进行了全面的误差分析。给出了平滑解和在 $t = 0$ 处具有奇异性的解的误差估计，这种类型对于涉及分数时间导数的方程来说是典型的。