## Abstract

Finite-difference based approaches are common for approximating the Caputo fractional derivative. Often, these methods lead to a reduction of the convergence rate that depends on the fractional order. In this note, we approximate the expressions in the fractional derivative components using a separate quadrature rule for the integral and a separate discretization of the derivative in the integrand. By this approach, the error terms from the Newton–Cotes quadrature and the differentiation are isolated and it is possible to conclude that the order dependent error is inevitable when the end points of the interval are included in the quadrature. Furthermore, we show experimentally that the theoretical findings carries over to quadrature rules without the end points included. Finally we show how to increase accuracy for smooth functions, and compensate for the order dependent loss.

Original language | English |
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Article number | 17 |

Journal | BIT Numerical Mathematics |

Volume | 63 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2023 |

## Keywords

- Closed quadrature rules
- Finite-differences
- Fractional derivative
- High-order numerical approximation
- Summation-by-parts operators

## ASJC Scopus subject areas

- Software
- Computer Networks and Communications
- Computational Mathematics
- Applied Mathematics