A Novel Theoretical Investigation of the Abu-Shady–Kaabar Fractional Derivative as a Modeling Tool for Science and Engineering

A newly proposed generalized formulation of the fractional derivative, known as Abu-Shady–Kaabar fractional derivative, is investigated for solving fractional differential equations in a simple way. Novel results on this generalized definition is proposed and verified, which complete the theory introduced so far. In particular, the chain rule, some important properties derived from the mean value theorem, and the derivation of the inverse function are established in this context. Finally, we apply the results obtained to the derivation of the implicitly defined and parametrically defined functions. Likewise, we study a version of the fixed point theorem for α-differentiable functions. We include some examples that illustrate these applications. The obtained results of our proposed definition can provide a suitable modeling guide to study many problems in mathematical physics, soliton theory, nonlinear science, and engineering.


Introduction
Fractional calculus is theoretically considered as a natural extension of classical differential calculus, which has attracted many researchers, both from a more theoretical point of view and for its diverse applications in sciences and engineering. Thus, from a more theoretical perspective, various definitions of fractional derivatives have been initiated. Fractional definitions try to satisfy the usual properties of the classical derivative; however, the only property inherent in these definitions is the property of linearity. On the contrary, some of the drawbacks that these derivatives present can be located in the following: More information on this definition of fractional derivative can be found in [1,2].
The locally formulated fractional derivative is established through certain quotients of increments. In this sense, Khalil et al. [3] introduced a locally defined derivative, called conformable derivative. Some of the inconveniences that the previous fractional derivatives presented have been successfully solved via this definition. Thus, for example, the aforementioned rules for the derivation of products and quotients of two functions or the chain rule are properties that have been satisfied by the conformable derivative. The physical and geometric meaning of the derivative is studied in [4,5]. However, in [6], the author shows the disadvantages of using the conformable definition compared to Caputo's fractional derivative definition, to solve some fractional models.
Recently, Abu-Shady and Kaabar [7] introduced a new generalized formulation of the fractional derivative (GFFD) that allows to solve analytically in a simple way some fractional differential equations, whose results agree exactly with those obtained via the Caputo and Riemann-Liouville derivatives. Also, this new definition has advantages compared to the conformable derivative definition. In addition, the study in [7] has been recently extended to study some important special functions in the sense of GFFD which are essential for modeling phenomena [8].
The GFFD definition is very important in studying various phenomena in science and engineering due to the powerful applicability of this definition in investigating many fractional differential equations in a very simple direction of obtaining analytical solutions without the need for approximate numerical methods or complicated algorithms like other classical fractional definitions. This definition is a modified version of the conformable definition to overcome all issues and advantages associated with the conformable one.
Regarding the geometric behavior of GFFD, by following the previous research study concerning the fractional cords orthogonal trajectories in the sense of conformable definition [5], GFFD can be similarly applied to the same example to interpret its geometrical meaning in more details.
One of the limitations of GFFD is that GFFD is locally defined derivative, and some future works are needed to proposed nonlocal formulation of GFFD in order to preserve the nonlocality property of fractional calculus. However, nonlocal definitions come with many associated challenges while working on solving fractional differential equations. Therefore, the future studies will work on overcoming all these challenges.
The work is constructed as follows: The GFFD and its main properties are presented in Section 2. New results on generalized α-differential functions are proposed in Section 3 to complete the study carried out in [7]. Some interesting applications of the results obtained on generalized α-differentiable functions are presented in Section 4. In particular, illustrative examples of the derivation of implicitly defined functions, of parametrically defined functions and of the application of the fixed point theorem for generalized α -differentiable functions are included. Some conclusions are drawn in Section 5.

New Results on Generalized α -Differentiable Functions
In this section, we establish important results that complete the theory of generalized α-differentiable functions, introduced in [7].
Proof. Since Then, Hence, f is continuous at t 0 Proof. We prove the result following a standard limit approach. First, if the function g is constant in a neighborhood of a > 0 then D GFFD ½ f ∘ gðtÞ = 0. If g not is constant in a neighborhood of a > 0, we can find a t 0 > 0 such that gðt 1 Þ ≠ gðt 2 Þ for any t 1 , t 2 ∈ ða − t 0 , a + t 0 Þ. Now, since g is continuous at a, for ε sufficiently small, we have Making in the first factor, so we have from here Remark 7. Using the fact that differentiability implies generalized α-differentiability and assuming gðtÞ > 0, Equation (6) can be written as Theorem 8 (Extended mean value theorem for generalized α -differentiable functions) [5]. Let a > 0, α ∈ ð0,1, and f , g : ½a, b ⟶ R be functions satisfying Then, ∃c ∈ ða, bÞ, ∋ Proof. Consider the function Since F is continuous on ½a, b, generalized α − DF on ð a, bÞ, and FðaÞ = FðbÞ = 0, then by Theorem 3, ∃c ∈ ða, bÞ such that D GF FD FðcÞ = 0. Using the linearity of D GFFD and the fact that the generalized α-derivative of a constant is zero, our result follows.
Theorem 12 (see [5]). Let a > 0, α ∈ ð0, 1, and f : ½a, b ⟶ R be a given function satisfying Proof. Following similar line of argument as given in the Theorem 10, there exists c between t 1 and t 2 with Therefore, f is strictly increasing on ½a, b, since t 1 and t 2 are arbitrary number of ½a, b Therefore, f is strictly decreasing on ½a, b, since t 1 and t 2 are arbitrary number of ½a, b.
Definition 13. Let I ⊂ ð0,∞Þ an open interval, α ∈ ð0,1, and f : I ⟶ R be we will say that f ∈ C α ðI, RÞ if the f is generalized α − DF on I and generalized α-derivative is continuous on I. Proof. Since

Applications
Some interesting applications of the results obtained on generalized α − DF functions are presented in this section.
Calculating the 1/3 -derivative in this equation, we obtain Taking t = 8 and gð8Þ = 1 in the equation above, we have Finally, the generalized 1/3-derivative is given by Taking p 0 = 1, to obtain this precision, 54 iterations are required. Also, note that since the generalized 1/2-derivative D GFFD f ðtÞ is negative, the successive approximations oscillate around the fixed point.

Conclusions
Novel results regarding the Abu-Shady-Kaabar fractional derivative have been investigated in this study which are extensions of the previous research study's results in [7]. In particular, some important properties of the generalized fractional derivative have been accomplished, such as the chain rule, some consequences of the mean value theorem, and the derivation of the inverse function. It is verifiable with the fact that these newly obtained results are considered as a natural extension of the classical differential calculus. The potential of this new definition of fractional derivative, both from a theoretical point of view and due to its applications, is evident through the developments and illustrative examples included in the previous section. This research can definitely open a new path for more related future works in which the results of classical mathematical analysis are extended in the sense of this new definition of fractional derivative. This definition will be applied further in studying various partial differential equations such as Schrödinger equation and Wazwaz-Benjamin-Bona-Mahony equation to study some solutions that are important in soliton theory and many other interesting research topics. Some specific examples of studies that can be further studied in the sense of GFFD are the Klein-Fock-Gordon equation via the Kudryashovexpansion method [9], the systems of fractional-order partial differential equations via the Laplace optimized decomposition technique [10], and the noninteger fractional-order hepatitis B model [11], by comparing the previous results in the senses of conformable and Caputo definitions with new results using GFFD. Numerical experiments with error analysis including comparison between conformable derivative and our definition including CPU time in the graphical representations in the sense of our proposed definition will be conducted in our future studies. In addition, in our future study, all algorithms and/ or pseudo-codes will be provided for the solutions' steps using one of the common software packages such as MAPLE and MATHEMATICA.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no competing interests.