We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency dependence power law. First we prove 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.
翻译:我们用一个非局部时间分断的术语来研究波形方程式,该方程式模拟以频率依赖力法为特征的声速减速效应。 首先,我们证明这个方程式存在独特的解决办法,特别注意分数衍生物的处理。 然后,我们根据空间的有限元素法以及分变二次和第二次中心时间差异的组合,得出一个明确的时间阶计划。 我们对混合时间分解进行完全错误分析,反过来又对全空时间分解的方程式进行完全错误分析。对单数为$t=0的平滑解决方案和单数为单数的解决方案都给出了错误估计,这种单数是涉及分数时间降解的方程式的典型类型。为了支持错误分析,我们提出了若干数字结果。