On fractional differential equations, dimensional analysis, and the double gamma function

TL;DR

This paper introduces a dimensional regularization for Caputo fractional derivatives to ensure physical consistency and demonstrates solutions involving the double gamma function, comparing them with standard approaches.

Contribution

It proposes a novel dimensional regularization method for Caputo derivatives and shows how to express solutions using the double gamma function.

Findings

Dimensional regularization ensures physical consistency in FDEs.

Solutions can be expressed using the double gamma function.

Comparison with standard Caputo derivatives highlights differences.

Abstract

In this paper we discuss some issues that arise in the process of writing a fractional differential equation (FDE) by replacing an integer order derivative by a fractional order derivative in a given differential equation. To address these issues, we propose a dimensional regularization of the Caputo fractional derivative, ensuring consistency in physical dimensions. Then we solve some FDEs using this proposed dimensional regularization. We show that the solutions of these FDEs are most conveniently written using the double gamma function. We also compare these solutions with those from equations involving the standard Caputo fractional derivative.