On fractional differential equations, dimensional analysis, and the double gamma function
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.
Cite
@article{arxiv.2506.00026,
title = {On fractional differential equations, dimensional analysis, and the double gamma function},
author = {J. Vaz and E. Capelas de Oliveira},
journal= {arXiv preprint arXiv:2506.00026},
year = {2025}
}
Comments
Preprint version. The Version of Record of this article is published in Nonlinear Dynamics, and is available online at https://doi.org/10.1007/s11071-025-11386-8