An introduction to fractional calculus: Numerical methods and application to HF dielectric response

TL;DR

This paper introduces fractional calculus concepts, explores its application to classical electrodynamics, especially high frequency dielectric response, and reviews numerical methods for fractional integration to aid modeling in electromagnetics.

Contribution

It provides a comprehensive overview of fractional calculus, links it to dielectric response, and reviews numerical techniques for practical modeling in electromagnetics.

Findings

Fractional calculus offers a natural framework for non-local phenomena.

Numerical methods for fractional integration enable practical modeling.

Application to high frequency dielectric response demonstrates real-world relevance.

Abstract

The aim of this work is to introduce the main concepts of Fractional Calculus, followed by one of its application to classical electrodynamics, illustrating how non-locality can be interpreted naturally in a fractional scenario. In particular, a result relating fractional dynamics to high frequency dielectric response is used as motivation. In addition to the theoretical discussion, a comprehensive review of two numerical procedures for fractional integration is carried out, allowing one immediately to build numerical models applied to high frequency electromagnetics and correlated fields.