Conformable derivative was first introduced by R. Khalil et al. [J. Comput. Appl. Math. 264, 65--70 (2014; Zbl 1297.26013)]. This type of derivative is local and limit-based derivative. I am wondering about the physical significance and geometrical interpretation of the conformable derivatives in comparison to the other non-local fractional derivatives. Please discuss that!!
References:
Khalil, R; Al Horani, M; Yousef, A; Sababheh, M, A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70, (2014) · Zbl 1297.26013 · doi:10.1016/j.cam.2014.01.002
see
(1)https://booksc.xyz/book/1419699/cb4979
(2)https://booksc.xyz/book/64032530/0f867a
It is noticed that the conformable derivative is a special case of directional derivative so the conformable fractional derivative is the velocity in a special direction
Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term \(t^{1-\alpha }\) in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD.
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