# A fast method for variable-order space-fractional diffusion equations

## TL;DR

This paper introduces a fast, efficient numerical method for solving variable-order space-fractional diffusion equations, overcoming structural challenges in the stiffness matrix and significantly reducing computational complexity.

## Contribution

The paper presents a novel divided-and-conquer collocation method that approximates the stiffness matrix with a sum of Toeplitz matrices, improving computational efficiency for variable-order fractional PDEs.

## Key findings

- Achieves $O(kN ext{log}^2 N)$ memory usage.
- Achieves $O(k N ext{log}^3 N)$ computational complexity.
- Demonstrates effectiveness through numerical experiments.

## Abstract

We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical approximation does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires $O(kN\log^2 N)$ memory and $O(k N\log^3 N)$ computational complexity with $N$ and $k$ being the numbers of unknowns and the approximants, respectively. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.