# Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative

## TL;DR

This paper develops fast, memory-efficient algorithms for computing Riemann-Liouville fractional derivatives using convolution quadrature, enabling efficient solutions to fractional Fokker-Planck equations with improved accuracy and reduced computational costs.

## Contribution

The paper introduces novel fast algorithms for fractional derivatives that do not assume solution regularity, significantly reducing computational time and memory for fractional PDEs.

## Key findings

- Algorithms achieve first- and second-order accuracy in time.
- Numerical examples confirm convergence and efficiency.
- Memory and computational costs are substantially decreased.

## Abstract

Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in [Nie, Sun and Deng, arXiv: 1811.04723], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative; and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides the fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods; the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply the fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first- and second-order accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of the fast algorithms.