We introduce a method which provides accurate numerical solutions to fractional-in-time partial differential equations posed on $[0,T] \times \Omega$ with $\Omega \subset \mathbb{R}^d$ without the excessive memory requirements associated with the nonlocal fractional derivative operator. Our approach combines recent advances in the development and utilization of multivariate sparse spectral methods as well as fast methods for the computation of Gauss quadrature nodes with recursive non-classical methods for the Caputo fractional derivative of general fractional order $\alpha > 0$. An attractive feature of the method is that it has minimal theoretical overhead when using it on any domain $\Omega$ on which an orthogonal polynomial basis is already available. We discuss the memory requirements of the method, present several numerical experiments demonstrating the method's performance in solving time-fractional PDEs on intervals, triangles and disks and derive error bounds which suggest sensible convergence strategies. As an important model problem for this approach we consider a type of wave equation with time-fractional dampening related to acoustic waves in viscoelastic media with applications in the physics of medical ultrasound and outline future research steps required to use such methods for the reverse problem of image reconstruction from sensor data.

翻译：我们提出一种方法，能够为定义在[0,T] × Ω（其中Ω ⊂ ℝ^d）上的时间分数阶偏微分方程提供精确数值解，同时避免与非局部分数阶导数算子相关的过多内存需求。该方法结合了多变量稀疏谱方法的最新进展、高斯求积节点的快速计算方法，以及针对一般分数阶α > 0的Caputo分数阶导数的递归非经典方法。该方法的一个吸引人之处在于，当将其应用于任何已有正交多项式基的域Ω时，其理论开销极小。我们讨论了该方法的内存需求，通过多个数值实验展示了该方法在区间、三角形和圆盘上求解时间分数阶偏微分方程的性能，并推导了误差界，从而提出了合理的收敛策略。作为该方法的一个重要模型问题，我们考虑了一类与黏弹性介质中的声波相关、具有时间分数阶阻尼的波动方程，该方程在医学超声物理学中有应用，并概述了未来研究步骤，以便将此方法应用于从传感器数据进行图像重建的反问题。