A New Fractional Derivative with Classical Properties

TL;DR

This paper introduces a new fractional derivative that retains classical calculus properties, generalizes polynomial derivatives, and aligns with the classical derivative when alpha equals one, offering a natural limit-based generalization.

Contribution

The paper proposes a novel fractional derivative definition that preserves classical properties and extends the concept of derivatives to non-integer orders using a limit approach.

Findings

The new derivative obeys linearity, product, quotient, power, and chain rules.

It satisfies Rolle's Theorem and the Mean Value Theorem.

For alpha=1, it coincides with the classical derivative.

Abstract

We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value Theorem. The definition, \[ D^\alpha (f)(t) = \lim_{\epsilon \rightarrow 0} \frac{f(te^{\epsilon t^{-\alpha}}) - f(t)}{\epsilon}, \] is the most natural generalization that uses the limit approach. For $0\leq \alpha < 1$, it generalizes the classical calculus properties of polynomials. Furthermore, if $\alpha = 1$, the definition is equivalent to the classical definition of the first order derivative of the function $f$. Furthermore, it is noted that there are $\alpha-$differentiable functions which are not differentiable.