We develop a fast indirect collocation method for a two-sided variable-order space-fractional diffusion equation, which models, e.g., the superdiffusive transport of solute in a heterogeneous porous medium. Due to the impact of the variable fractional order, the stiffness matrix loses the Toeplitz structure that is common in the context of constant-order sFDEs. Consequently, the discrete fast Fourier transform technique or fast convolution quadrature method used in the development of fast methods for constant-order fractional difusion equations no longer apply. We approximate the stiffness matrix by a combination of Toeplitz-like matrices via an entrywise expansion. We prove that the approximated system are asymptotically consistent with the original problem with only $O(NlogN)$ memory and $O\left(N{log}^{2}N\right)$ operations are required per iteration for the fast solvers. We prove that the fast numerical solution of the approximated system has the same order of accuracy as that of the underlying collocation method does, without any artificial regularity assumption of the true solution. Numerical experiments are presented to demonstrate the utility of the method.

我们为双面可变阶空间分数分数扩散方程开发了一种快速间接配置方法，该方法可模拟例如溶质在非均质多孔介质中的超扩散传输。由于可变分数阶的影响，刚度矩阵失去了在恒定阶sFDE中常见的Toeplitz结构。因此，用于定阶分数阶扩散方程的快速方法开发中使用的离散快速傅里叶变换技术或快速卷积正交方法不再适用。我们通过进入展开将Toeplitz样矩阵组合起来来近似刚度矩阵。我们证明近似系统与原始问题渐近一致$Ø（ñ日志ñ）$ 记忆和 $Ø（ñ{日志}^2ñ）$快速求解器的每次迭代都需要操作。我们证明了近似系统的快速数值解具有与基础配置方法相同的精确度，而没有任何人为的正则性假设。数值实验表明了该方法的实用性。