In my oppinion, as we see from the definition of the conformable fractional derivative, only the multiplication of the classical derivative by x ^ {1-a} can not bring any novelty in the study of real functions.
Also, the use of the term of conformable fractional derivative is unfortunate and creates confusion because there is no connection between this concept and Riemann-Liouville fractional derivative or Caputo fractional derivative. In fact, the conformal fractional derivative of a given function f is nothing but the derivative of the function f with respect cu other function, namely g(x)=x^{α}, in our case. If we put u=x^{α}, then it esay to see that
((df)/(dx))=((df)/(du))((du)/(dx))=α((df)/(du))x^{α-1},
so that
((df)/(du))=(1/α)((df)/(dx))x^{1-α}=(1/α)D^{α}f,
where D^{α} is the conformable derivative.
The same question for Katugampola fractional derivative?.
Dear,
Try to read this artical and references therein:
https://www.sciencedirect.com/science/article/pii/S0377042714004622
Dear Majeed,@
I have seen almost everything that has been published about this conformable fractional derivative.
What is novelty?
Regards,
Vasile
Conformable derivative is not a fractional derivative, it is an extension to the classical ones.
Dear friend,
Have a look on following research item, it may be helpful for you.
Dear Sandeep Pandurang Bhairat,
Please, read Chapter 2 (paragraph 2.5) in [Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations, volume 204 of North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 2006] puting g(x)=x^ρ.
In fact, the idea a such derivative is not so new. See: V.V. Kobelev, On a Leibnitz-type fractional derivative, https://arxiv.org/abs/1202.2714
Recently, Khalil et al. (2014) introduced a new simple well-behaved definition of the fractional derivative called conformable fractional derivative. This new fractional derivative definition has governed much attention in recent months. For instance Abdeljawad (2015) provide fractional versions of the chain rule, exponential functions, Gronwalls inequality, integration by parts, Chung (2015) used the conformable fractional derivative and integral to discuss fractional Newtonian mechanics and Rezazdeh et al. investigated stability of linear conformable fractional systems from the point of view of control (Rezazadeh et al. 2017). It is worth noting that the conformable fractional derivative dont have a physical meaning as the Caputo or Riemann–Liouville derivatives.
The conformable fractional derivative has a wide applications which can be seen from the following selected papers:
https://booksc.xyz/book/47010726/c201b9
https://booksc.xyz/book/64032530/0f867a
https://booksc.xyz/book/67387259/dd2687
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