A New Fractional Derivative with Classical Properties

By Udita N. Katugampola. 2014

179 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/79bfdf3ecd4e1b41a643ca8948126d4ab6786da2

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In Control and Systems Engineering

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

A critical analysis of the conformable derivative (2018, background): “We have demonstrated in Theorem 2 that there is no function α -differentiable in the sense of Katugampola [13] that is not differentiable.”
Newly Defined Conformable Derivatives (2015, background): “or in [14] as Df(t) := lim ε→0 f(te −α )− f(t) ε , Df(0) = lim t→0+ Df(t), provided the limits exist; note that if f is fully differentiable at t, then in either case we have Df(t) = t1−αf ′(t), (1.”
General conformable fractional derivative and its physical interpretation (2017, background): “[19] and [20], these derivatives do not obey product rule, quotient rule, chain rule for derivative operations.”
A new truncated $M$-fractional derivative type unifying some fractional derivative types with classical properties (2017, background): “In the same paper, Katugampola [8] by means of a truncated exponential function, that is, ke , proposed another generalized fractional derivative, given by”
The flaw in the conformable calculus: It is conformable because it is not fractional (2018, background): “Analogous remarks apply to the statements and proofs in [3] and [4].”