# Efficient high order algorithms for fractional integrals and fractional differential equations

## TL;DR

This paper introduces an efficient, high-order algorithm for fractional integrals and differential equations using Runge--Kutta convolution quadrature, with rigorous error estimates and demonstrated numerical performance.

## Contribution

It presents a novel integral representation and quadrature method for fractional integrals, enabling high-order, efficient solutions with rigorous error analysis.

## Key findings

- Algorithm achieves high accuracy with low memory and computational cost.
- Numerical results validate the efficiency and accuracy of the method.
- Application to fractional diffusion equations demonstrates practical utility.

## Abstract

We propose an efficient algorithm for the approximation of fractional integrals by using Runge--Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The resulting method is easy to implement, allows for high order, relies on rigorous error estimates and its performance in terms of memory and computational cost is among the best to date. Several numerical results illustrate the method and we describe how to apply the new algorithm to solve fractional diffusion equations. For a class of fractional diffusion equations we give the error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge--Kutta based convolution quadrature in time.