On the nature of the conformable derivative and its applications to physics

TL;DR

This paper demonstrates that conformable derivatives are mathematically equivalent to a simple variable change, but explores their potential value in physics and engineering, especially in quantum mechanics.

Contribution

It clarifies the mathematical nature of conformable derivatives and investigates their applications and interpretations in physics and engineering contexts.

Findings

Conformable derivatives are equivalent to a change of variables in differential functions.

Exploration of conformable derivatives in linear differential equations and Sturm-Liouville systems.

Application of conformable derivatives to quantum mechanics and physical models.

Abstract

The purpose of this work is to show that the Khalil and Katagampoula conformable derivatives are equivalent to the simple change of variables $x$ $\rightarrow $ $x^{\alpha }/\alpha ,$ where $\alpha $ is the order of the derivative operator, when applied to differential functions. Although this means no \textquotedblleft new mathematics\textquotedblright\ is obtained by working with these derivatives, it is a second purpose of this work to argue that there is still significant value in exploring the mathematics and physical applications of these derivatives. This work considers linear differential equations, self-adjointness, Sturm-Liouville systems, and integral transforms. A third purpose of this work is to contribute to the physical interpretation when these derivatives are applied to physics and engineering. Quantum mechanics serves as the primary backdrop for this development.