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：本文提出一种方法，可为定义在 $[0,T] \times \Omega$（其中 $\Omega \subset \mathbb{R}^d$）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度存储需求。该方法结合了多元稀疏谱方法的最新进展，以及利用递归非经典方法计算Caputo分数阶导数（一般阶数 $\alpha > 0$）的高斯求积节点的快速算法。该方法的一个显著优势在于：当应用于任何已存在正交多项式基的域 $\Omega$ 时，其理论开销极小。我们讨论了该方法的存储需求，通过多个数值实验展示了其在区间、三角形和圆盘上求解时间分数阶偏微分方程的性能，并推导了误差界以提出合理的收敛策略。作为该方法的典型模型问题，我们研究了一种与粘弹性介质中的声波相关（应用于医学超声物理）的含时间分数阶阻尼波动方程，并概述了未来需开展的研究步骤，以将该方法用于传感器数据图像重建的反问题。