# Extending discrete exterior calculus to a fractional derivative

## TL;DR

This paper extends discrete exterior calculus to fractional derivatives, enabling more accurate numerical solutions for fractional PDEs involving non-local interactions, by defining a fractional discrete derivative based on DEC principles.

## Contribution

It introduces a Caputo-like fractional discrete derivative within DEC, bridging the gap between fractional calculus and geometric discretization methods.

## Key findings

- The fractional discrete derivative retains key properties of the continuous derivative.
- Numerical experiments demonstrate the effectiveness of the proposed discretization.
- The method provides a new computational tool for fractional PDEs.

## Abstract

Fractional partial differential equations (FDEs) are used to describe phenomena that involve a "non-local" or "long-range" interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the dense matrices arising from standard discretization procedures. In this paper, we begin to extend the well-established computational toolkit of Discrete Exterior Calculus (DEC) to the fractional setting, focusing on proper discretization of the fractional derivative. We define a Caputo-like fractional discrete derivative, in terms of the standard discrete exterior derivative operator from DEC, weighted by a measure of distance between $p$-simplices in a simplicial complex. We discuss key theoretical properties of the fractional discrete derivative and compare it to the continuous fractional derivative via a series of numerical experiments.