Fractional Calculus, conformable derivative, non conformable derivative, generalized derivative

Fractional calculus concerns the generalization of differentiation and integration to non-integer (fractional) orders. The subject has a long mathematical history being discussed for the first time already in the correspondence of Leibniz with L’Hopital when this replied "What does $\frac{d^n}{d{x}^n}f\left(x\right)$ mean if n=$\frac{1}{2}$ ?" in September 30 of 1695. Over the centuries many mathematicians have built up a large body of mathematical knowledge on fractional integrals and derivatives. Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. One reason could be that, until recently, the basic facts were not readily accessible even in the mathematical literature (see¹). The nature of many systems makes that they can be more precisely modeled using fractional differential equations. The differentiation and integration of arbitrary orders have found applications in diverse fields of science and engineering like viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals (see^1-3).

The history of differential operators from Newton to Caputo, both local and global, is given in⁴ (Chapter 1). Here is the definition of a local derivative with a new parameter, which has a large number of applica- tions. More importantly, section 1.4 concludes: “Therefore, we can conclude that both the Riemann-Liouville operator and the Riemann-Liouville operator Caputo are not derivatives, and therefore they are not fractional derivatives, but fractional operators”. We are of in agreement with the result⁵ that says “the local fractional operator is not a fractional derivative”(page 24). As mentioned above, these tools are new and have demonstrated their potential and usefulness in solving phenomena and process modeling problems in various fields of science and technology (see⁶). Many different types of fractional operators have been proposed in the literature, here we show that several of these different notions of derivatives can be considered particular cases of our definition and, even more relevant, that it is possible to establish a direct relationship between derivatives global (classical) and local, the latter not very accepted by the mathematical community, under two arguments: its local character and compliance with the Leibniz Rule. In this note we present the recent development of the so-called Non-Integer Order Local Calculus, which is the correct name (sometimes we use the name Generalized Calculus, although it does not illustrate the concept well). To facilitate the understanding of the scope of our objective, we present the best known definitions of differential and integral local operators (for more details you can consult,^7,8 Without much difficulty, we can extend these definitions, for any higher order. We assume that the reader is familiar with the classical Calculus, so we will not present it.

Local fractional calculus (is also called Fractal calculus) was first introduced by Kolwankar and Gangal, although there were some attempts in the 1960s, this is the first formal definition of a local operator that generalizes the classical derivative. It is explain the behavior of continuous but nowhere differentiable function. They proposed particular notation that they had used in their publication for the local fractional derivative of a function defined on fractal sets.^9-11 So we have

Definition 1 If, for a function $f:\left[0,1\right]\to R$ , the limit

$D^{q}f\left(y\right)=\underset{x\to y}{\text{lim}}\frac{d^q\left(f\left(x\right)-f\left(y\right)\right)}{d{(x-y)}^q},$   (1)

exists and is finite, then we say that the local fractional derivative (LFD) of order q, at x = y, exists.

To understand the fractal behavior of functions, Parvate and Gangal (see)¹² introduce the fractal derivative as follows:

$x_0{D}_x^{\alpha }f\left(y\right)=\frac{d^{\alpha }f\left(x\right)}{d{x}^{\alpha }}\left(x_0\right)=F-\underset{x\to x_0}{\text{lim}}\frac{f\left(x\right)-f\left(x_0\right)}{S_F^{\alpha }\left(x\right)-S_F^{\alpha }\left(x_0\right)}$   (2)

where the right hand side is the notion of the limit by the points of the fractal set F.

Definition 2 Let  be an arbitrary but fixed real number. The integral staircase function $S_F^{\alpha }\left(x\right)$ of order α for a set is given by:

$S_F^{\alpha }\left(x\right)=\{\begin{array}{lll}{\gamma }^{\alpha }\left[F,a,x\right]\hfill & si\hfill & x\ge a\hfill \\ \hfill & \hfill & \hfill \\ -{\gamma }^{\alpha }\left[F,a,x\right]\hfill & si\hfill & x<a\hfill \end{array}$   (3)

and the mass function is defined in this way.

Definition 3 The mass function ${\gamma }^{\alpha }\left[F,a,b\right]$ can written as (see^13,14):

${\gamma }^{\alpha }\left[F,a,b\right]=\underset{\delta \to 0}{\text{lim}}{\gamma }_{\delta }^{\alpha }\left[F,a,b\right]=\frac{{\left(b-a\right)}^{\alpha }}{\text{Γ}\left(1+\alpha \right)}$ .  (4)

Another version can be found at:¹⁵

$x_0{D}_x^{\alpha }f\left(y\right)=\frac{d^{\alpha }f\left(x\right)}{d{x}^{\alpha }}\left(x_0\right)=\underset{x\to x_0^{\sigma }}{\text{lim}}{D}_{y,-\sigma }^{\alpha }\left[\sigma \left(f\left(x\right)-f(x_0\right)\left(x\right)\right]$ ,  (5)

with $\sigma =\pm$ and $D_{y,-\sigma }^{\alpha }$ is the Riemann-Liouville derivative.

In¹⁶ we have the following notion:

$x_0{D}_x^{\alpha }f\left(y\right)=\frac{d^{\alpha }f\left(x\right)}{d{x}^{\alpha }}\left(x_0\right)=\underset{x\to x_0}{\text{lim}}\frac{f\left(x\right)-f\left(x_0\right)}{x^{\alpha }-x_0^{\alpha }}$ ,  (6)

obtained from (??) under assumption $x^{\alpha }-x_0^{\alpha }={(x-x_0)}^{\alpha }$ .

He gave a new fractal derivative in theis way:¹⁷

$x_0{D}_x^{\alpha }f\left(y\right)=\frac{d^{\alpha }f\left(x\right)}{d{x}^{\alpha }}\left(x_0\right)=\underset{\text{Δ}x\to L_0}{\text{lim}}\frac{f\left(x\right)-f\left(x_0\right)}{K{L}_0^{\alpha }}$ .  (7)

Taking into account.

$H^{\alpha }\left(F\cap \left(x,x_0\right)\right)=\left(x-x_0^{\alpha }\right)=\frac{K}{\text{Γ}\left(1+\alpha \right)}{L}_0^{\alpha }$ .

Yjis is the unified notation of.¹⁸ In this address we have another definition,^19,20 as follows:

$D^{\alpha }f\left(x\right)=\frac{d^{\alpha }f\left(x\right)}{d{x}^{\alpha }}\left(x_0\right)=\underset{x\to x_0}{\text{lim}}\frac{{\text{Δ}}^{\alpha }\left[f\left(x\right)-f\left(x_0\right)\right]}{{(x-x_0)}^{\alpha }}$ ,  (8)

${(x-x_0)}^{\alpha }$  is a measure fractal²⁰ and ${\text{Δ}}^{\alpha }\left[f\left(x\right)-f\left(x_0\right)\right]\cong \text{Γ}\left(1+\alpha \right)\text{Δ}\left[f\left(x\right)-f\left(x_0\right)\right]$ . In [68] we have:

$D^{\alpha }f\left(x\right)=\frac{d^{\alpha }f\left(x\right)}{d{x}^{\alpha }}\left(x_0\right)=\underset{x\to x_0}{\text{lim}}\frac{f\left(x\right)-f\left(x_0\right)}{{(x-x_0)}^{\alpha }}$ .  (9)

All these results, although they do not exactly coincide with the direction of our work, we present them so that readers have a more complete picture and because they have become relevant again in recent years.

3  Post Kahlil derivative

In²¹ a definition of local derivative is presented, which opens a new direction of work, which is what we intend to illustrate here.

So they presented the following definition(see also²²).

Thus, for a function $f:\left(0,\infty \right)\to R$ the conformable derivative of order $0<\alpha \le 1$ of f at $t>0$ was defined by.

$T_{\alpha }f\left(t\right)=li{m}_{\epsilon \to 0}\frac{f\left(t+\epsilon t^{1-\alpha }\right)-f\left(t\right)}{ϵ}$ ,  (10)

and the fractional derivative at 0 is defined as $T_{\alpha }f\left(0\right)=li{m}_{t\to 0}{T}_{\alpha }f\left(t\right)$ .

In a work from the same year (cf.)⁵⁸ another conformable derivative is defined in a very similar way. Let f be a function of $\left(0,\infty \right)\to ℝ$ , $t>0$ define the derivative of order α with $0<\alpha <1$ as the expression $D^{\alpha }f\left(t\right)=li{m}_{\epsilon \to 0}\frac{f\left(t{e}^{\epsilon t^{-\alpha }}\right)-f\left(t\right)}{\epsilon }$ , of course, if $D^{\alpha }f\left(t\right)$ exists at some $\left(0,a\right)$ with $a>0$ then defines the derivative of order α at 0 as $D^{\alpha }f\left(0\right)=li{m}_{t\to 0}{D}^{\alpha }f\left(t\right)$ .

⁸introduces a new twist when it defines a general derivative as follows, $f:ℝ\to ℝ$ is a function, α a real number, the derivative of fractional order can be thought of as $f^{\alpha }\left(t\right)=li{m}_{\epsilon \to 0}\frac{f^{\alpha }\left(t+\epsilon \right)-f\left(t\right)}{{(t+\epsilon )}^{\alpha }-t^{\alpha }}$ .

In 2018 we introduced a new local derivative, with a very distinctive property: when $\alpha \to 1$ we do not get the ordinary derivative. We call this derivative non-conformable, to distinguish it from the previous known ones, since when $\alpha \to 1$ the slope of the tangent line to the curve at the point is not preserved.

Be $\alpha \in \left(0,1\right]$ and define a continuous function $f:\left[t_0,+\infty \right)\to ℝ$ .

First, let’s remember the definition of ${}_1{N}^{\alpha }f\left(t\right)$ , a non conformable fractional derivative of a function in a point t defined in²³ and that is the basis of our results, that are close resemblance of those found in classical qualitative theory.

Definition 4 Given a function $f:\left[t_0,+\infty \right)\to ℝ$ , $t_0>0$ . Then the N-derivative of f of order α is defined by ${ }_1{N}^{\alpha }f\left(t\right)=\underset{\epsilon \to 0}{lim}\frac{f\left(t+\epsilon e^{t^{-\alpha }}\right)-f\left(t\right)}{\epsilon }$ for all $t>0$ , $\alpha \in \left(0,1\right)$ . If f is α - differentiable in some $\left(0,a\right)$ , and $\underset{t\to 0^{+}}{lim}{N}_1^{\left(\alpha \right)}f\left(t\right)$  exists, then define ${ }_1{N}^{\alpha }f\left(0\right)=\underset{t\to 0^{+}}{lim}{N}_1^{\left(\alpha \right)}f\left(t\right)$ .

If the above derivative of the function $x\left(t\right)$ of order α exists and is finite in $\left(t_0,\infty \right)$ , we will say that $x\left(t\right)$ is $N_1$ - differentiable in $I=\left(t_0,\infty \right)$ .

Remark 5 The use in Definition 1 of the limit of a certain incremental quotient, instead of the integral used in the classical definitions of fractional derivatives, allows us to give the following interpretation of the N-derivative. Suppose that the point moves in a straight line in ${ℝ}_{+}$ . For the moments $t_1=t$ and $t_2=t+h{e}^{t^{-\alpha }}$ where $h>0$ and $\alpha \in \left(0,1\right]$ and we denote $S\left(t_1\right)$ and $S\left(t_2\right)$ the path traveled by point P at time $t_1$ and $t_2$ so we have $\frac{S\left(t_2\right)-S\left(t_1\right)}{t_2-t_1}=\frac{S\left(t+h{e}^{t^{-\alpha }}\right)-S\left(t\right)}{h{e}^{t^{-\alpha }}}$ this is the average $N_1$ - speed of point P over time $h{e}^{t^{-\alpha }}$ . Let’s consider $\underset{h\to 0}{Lim}\frac{S\left(t+h{e}^{t^{-\alpha }}\right)-S\left(t\right)}{h{e}^{t^{-\alpha }}}$ .

When $\alpha =1$ , this is the usual instantaneous velocity of a point P at any time $t>0$ . If $\alpha \in \left(0,1\right)$ this is the instantaneous q-speed of the point P for any $t>0$ . Therefore, the physical meaning of the N-derivative is the instantaneous q-change rate of the state vector of the considered mechanics or another nature of the system.

Remark 6 The $N_1$ - derivative solves almost all the insufficiencies that are indicated to the classical fractional derivatives. In particular we have the following result.

Theorem 7 (See²⁴) Let f and g be N-differentiable at a point $t>0$ and $\alpha \in \left(0,1\right]$ . Then.

1. $N_1^{\alpha }\left(af+bg\right)\left(t\right)=a_1{N}^{\alpha }\left(f\right)\left(t\right)+b_1{N}^{\alpha }\left(g\right)\left(t\right)$ .
2. ${}_1{N}^{\alpha }\left(t^p\right)=e^{t^{-\alpha }}p{t}^{p-1}, p\in ℝ$ .
3. ${}_1{N}^{\alpha }\left(\lambda \right)=0, \lambda \in ℝ$ .
4. ${}_1{N}^{\alpha }\left(fg\right)\left(t\right)=f_1{N}^{\alpha }\left(g\right)\left(t\right)+g_1{N}^{\alpha }\left(f\right)\left(t\right)$ .
5. ${}_1{N}^{\alpha }\left(\frac{f}{g}\right)\left(t\right)=\frac{g{N}_1^{\alpha }\left(f\right)\left(t\right)-f_1{N}^{\alpha }\left(g\right)\left(t\right)}{g^2\left(t\right)}$ .
6. If, in addition, f is differentiable then  $N_1^{\alpha }\left(f\right)=e^{t^{-\alpha }}{f}^{\prime }\left(t\right)$ .
7. Being f differentiable and $\alpha =n$ integer, we have $N_1^n\left(f\right)\left(t\right)=e^{t^{-n}}{f}^{\prime }\left(t\right)$ .

Remark 8 The relations a), c), d) and (e) are similar to the classical results mathematical analysis, these relationships are not established (or do not occur) for fractional derivatives of global character (see^1,2 and bibliography there). The relation c) is maintained for the fractional derivative of Caputo. Cases c), f) and g) are typical of this non conformable local fractional derivative.

Now we will present the equivalent result, for ${}_1{N}^{\alpha }$ , of the well-known chain rule of classic calculus and that is basic in the Second Method of Lyapunov, for the study of stability of perturbed motion.

Theorem 9 (See²⁴) Let $\alpha \in \left(0,1\right]$ , g N-differentiable at $t>0$ and f differentiable at $g\left(t\right)$ then ${ }_1{N}^{\alpha }\left(f\circ g\right)\left(t\right)=f^{\prime }{(g\left(t\right))}_1{N}^{\alpha }g\left(t\right)$ .

Definition 10 The non conformable fractional integral of order α is defined by the expression ${ }_1{J}_{t_0}^{\alpha }f\left(t\right)=\underset{t_0}{\overset{t}{{\displaystyle \int }}}\frac{f\left(s\right)}{e^{s^{-\alpha }}}ds$ .

The following statement is analogous to the one known from the Ordinary Calculus.

Theorem 11 Let f be $N_1$ -differentiable function in $\left(t_0,\infty \right)$ with $\alpha \in \left(0,1\right]$ . Then for all $t>t_0$ we have.

1. If f is differentiable ${}_1{J}_{t_0}^{\alpha }\left({}_1{N}^{\alpha }f\left(t\right)\right)=f\left(t\right)-f\left(t_0\right)$ .
2. ${}_1{N}^{\alpha }\left({}_1{J}_{t_0}^{\alpha }f\left(t\right)\right)=f\left(t\right)$ .

Proof: See²⁵

This derivative, and some variants, proved useful in various application problems (see^26-35).

4  The N-derivative

In³⁶ a generalized derivative was defined as follows (see also^37,38).

Definition 12  Given a function $\psi :\left[0,+\infty \right)\to ℝ$ . Then the N-derivative of $\psi$ of order α is defined by

$N_F^{\alpha }\psi \left(\tau \right)=\underset{\epsilon \to 0}{\text{lim}}\frac{\psi \left(\tau +\epsilon F\left(\tau ,\alpha \right)\right)-\psi \left(\tau \right)}{\epsilon }$    (11)

for all $\tau >0$ , $\alpha \in \left(0,1\right)$ being $F\left(\tau ,\alpha \right)$ is some function.

If $\psi$ is N-differentiable in some $\left(0,\alpha \right)$ , and $\underset{\tau \to 0^{+}}{\text{lim}}{N}_F^{\alpha }\psi \left(\tau \right)$ exists, then define $N_F^{\alpha }\psi \left(0\right)=\underset{\tau \to 0^{+}}{\text{lim}}{N}_F^{\alpha }\psi \left(\tau \right)$ , note that if $\psi$ is differentiable, then $N_F^{\alpha }\psi \left(\tau \right)=F\left(\tau ,\alpha \right){\psi }^{\prime }\left(\tau \right)$ where ${\psi }^{\prime }\left(\tau \right)$ is the ordinary derivative.

Examples. Let’s see some particular cases that provide us with new non-conforming derivatives. 

1. Mellin-Ross Function. In this case we have

$E_t\left(\alpha ,a\right)=t^{\alpha }{E}_{1,\alpha +1}\left(at\right)=t^{\alpha }{\displaystyle \sum }_{k=0}^{\infty }\frac{{(at)}^k}{\text{Γ}\left(\alpha +k+1\right)}$

with $E_{1,\alpha +1}(.)$ the Mittag-Leffler two-parameter function. So, we obtain $\underset{\alpha \to 1}{\text{lim}}{N}_{E_t\left(\alpha ,a\right)}^{\alpha }f\left(t\right)=f\text{'}\left(t\right)t{E}_{1,2}\left(at\right)$ , i.e., $N_{E_t\left(1,a\right)}^{1}f\left(t\right)=f\text{'}\left(t\right)t{\displaystyle \sum }_{k=0}^{\infty }\frac{{(at)}^k}{\text{Γ}\left(k+2\right)}$ .

2. Robotov’s Function. That is to say 

$R_{\alpha }\left(\beta ,t\right)=t^{\alpha }{\displaystyle \sum }_{k=0}^{\infty }\frac{{\beta }^k{t}^{k\left(\alpha +1\right)}}{\text{Γ}\left(1+\alpha \right)\left(k+1\right)}=t^{\alpha }{E}_{\alpha +1,\alpha +1}\left(\beta t^{\alpha +1}\right)$

like before, $E_{\alpha +1,\alpha +1}(.)$ is the Mittag-Leffler two-parameter function. Now, we obtain $\underset{\alpha \to 1}{\text{lim}}{N}_{R_{\alpha }\left(\beta ,t\right)}^{\alpha }f\left(t\right)=f\text{'}\left(t\right)t{E}_{2,2}\left(\beta t^2\right)$ and

$N_{R_1\left(\beta ,t\right)}^{1}f\left(t\right)=\frac{f\text{'}\left(t\right)t}{\text{Γ}\left(2\right)}{\displaystyle \sum }_{k=0}^{\infty }\frac{{\beta }^k{t}^{2k}}{\left(k+1\right)}$

3. Let $F\left(t,\alpha \right)=E_{1,1}\left(t^{-\alpha }\right)$ . In this case we obtain, from Definition 12, the derivative ${}_1{N}^{\alpha }f\left(t\right)$ defined in [18] (and [46]).
4. Be now $F\left(t,\alpha \right)=E_{1,1}{(t^{-\alpha })}_1$ , in this case we have $F\left(t,\alpha \right)=\frac{1}{t^{\alpha }}$ , a new derivative with a remarkable propertie ${\text{lim}}_{t\to \infty }{N}_1^{\alpha }f\left(t\right)=0$ , i.e., the derived N is annulled to infinity.
5. If we now take the development of function E to order 1, we have $E_{a,b}\left(t^{-\alpha }\right)=1+\frac{1}{t^{\alpha }}$ . Then ${\text{lim}}_{t\to \infty }{N}_F^{\alpha }f\left(t\right)={\text{lim}}_{t\to \infty }{}_1{N}^{\alpha }f\left(t\right)=f\text{'}\left(t\right)$ , in this case we have the classic derivative at infinit.

Remark 13 It is easy to check but tedious, following for example, that the general derivative fulfills properties very similar to those known from the classical calculus. As well as its most important consequences, among them the Chain Rule, of vital importance in many applications, among them the Second Method of Lyapunov.

Remark 14 The generalized derivative defined above is not fractional (as we noted above), but it does have a very desirable feature in applications, its dual dependency on both  and the kernel expression itself, with $0<\alpha leq1$ in ²¹ the conformal derivative is defined by putting $F\left(t,\alpha \right)=t^{1-\alpha }$ , while in²⁴ the nonconforming derivative is obtained with $F\left(t,\alpha \right)=e^{t^{-\alpha }}$ (see also ²⁵). This generalized derivative, in addition to the aforementioned cases, contains as particular cases practically all known local operators and has proved its utility in various applications, see, for example,^{23,30,32-35,39,40-52}

Remark 15 One of the characteristics of this generalized derivative is the fact that $N_F^{2\alpha }f\left(t\right)\ne N_F^{\alpha }\left(N_F^{\alpha }f\left(t\right)\right)$ , that is, it is necessary to indicate successive derivatives in the second way. Obviously, if $N_F^{2\alpha }f\left(t\right)\ne N_F^{\alpha }\left(N_F^{\alpha }f\left(t\right)\right)$ , the ordinary derivative is obtained.

Remark 16  From the above definition, it is not difficult to extend the order of the derivative for $0\le n-1<\alpha \le n$ by putting.

$N_F^{\alpha }h\left(\tau \right)=\underset{\epsilon \to 0}{\text{lim}}\frac{h^{\left(n-1\right)}\left(\tau +\epsilon F\left(\tau ,\alpha \right)\right)-h^{\left(n-1\right)}\left(\tau \right)}{\epsilon }$   (12)

If $h^{\left(n\right)}$ exists on some interval $I\subseteq ℝ$ , then we have $N_F^{\alpha }h\left(\tau \right)=F\left(\tau ,\alpha \right)h^{\left(n\right)}\left(\tau \right)$ , with $0\le n-1<\alpha \le n$ .

Slightly more recent, in³⁷ a notion of generalized fractional derivative is defined, which is general from two points of view:

1) Contains as particular cases, both conformable and non-conformable derivatives.

2) It is defined for any order $\alpha >0$ .

Given $s\in ℝ$ , we denote by $\text{[}s]$ the upper integer part of s, i.e., the smallest integer greater than or equal to s.

Definition 17 Given an interval $I\subseteq \left(0,\infty \right)$ , $f:I\to ℝ$ , $\alpha \in {ℝ}^{+}$ and a continuous function positive $T\left(t,\alpha \right)$ , the derivative $G_T^{\alpha }f$ of f of order α at the point $t\in I$ is defined by.

$G_T^{\alpha }f\left(t\right)=\underset{h\to 0}{\text{lim}}\frac{1}{h^a}{\displaystyle \sum }_{k=0}^a{(-1)}^k\left(\begin{array}{l}a\hfill \\ k\hfill \end{array}\right)f\left(t-khT\left(t,\alpha \right)\right)$   (13)

In 2018, a derivative operator is defined on the real line with a limit process as follows (se⁵³). For a given function p of two variables, the symbol $D_{p}f\left(t\right)$ defined by the limit $D_{p}f\left(t\right)={\text{lim}}_{\epsilon \to 0}\frac{f\left(p\left(t,\epsilon \right)\right)-f\left(t\right)}{\epsilon }$ , as long as the limit exists and is finite, it will be called the derivative p of f at t or the generalized derivative from f to t and, for brevity, we also say that f is p-differentiable in t. In the case that it is a closed interval, we define the p-derivative at the extremes as the respective side derivatives. Starting from this definition, the derivative of order of a function is constructed as the following limit:

$D_p^{\alpha }f\left(t\right)=\underset{\epsilon \to 0}{\text{lim}}\frac{f\left(p\left(t,\epsilon ,\alpha \right)\right)-f\left(t\right)}{\epsilon }, 0<\alpha <1$   (14)

where it is understood that in the case $\alpha =1$ we have the ordinary derivative. It is clear that if f is differentiable in t, then $D_p^{\alpha }f\left(t\right)=p_h\left(t,0,\alpha \right)f\text{'}\left(t\right), 0<\alpha <1$ . Note that there are no sign restrictions on the function p nor in its partial derivative $p_h\left(t,0,\alpha \right)$ .

There is an additional detail that we want to point out, in³⁶ the following is pointed out.

However, a new local derivative that violates Leibniz’s Rule can be constructed, so the violation of this rule cannot be a necessary condition for a given operator to be a fractional derivative, let’s go back to (11). It is clear that the violation of this rule does not depend (at least not only) on the incremental quotient, but on a factor that we can add to the increased function, from which the non-symmetry of the product rule would be obtained.

Taking into account⁵⁴ we can write from (11) the following derivative $\left(\alpha +\beta =1\right)$ :

$D{H}_{\beta }^{\alpha }f\left(t\right):=\underset{\epsilon \to 0}{\text{lim}}\frac{H\left(\epsilon ,\beta \right)f\left(t+\epsilon F\left(t,\alpha \right)\right)-f\left(t\right)}{\epsilon }$   (15)

with $H\left(\epsilon ,\beta \right)\to k$ if $\epsilon \to 0$ . In the case that $k\equiv 1$ , we can consider two simple cases: 

1. $H\left(\epsilon ,\beta \right)=1+\epsilon \beta$  as in⁵⁴ and so 

$D{L}_{\beta }^{\alpha }f\left(t\right):=\underset{\epsilon \to 0}{\text{lim}}\frac{\left(1+\epsilon \beta \right)f\left(t+\epsilon F\left(t,\alpha \right)\right)-f\left(t\right)}{\epsilon }$ .

If $F\left(t,\alpha \right)=e^{t^{-\alpha }}$ , that is, a generalization of the local fractional derivative presented in example 4 above. In this case we have:

$N{L}_2^{\alpha }f\left(t\right):=\underset{\epsilon \to 0}{\text{lim}}\frac{\left(1+\epsilon \beta \right)f\left(t+\epsilon e^{t^{-\alpha }}\right)-f\left(t\right)}{\epsilon }$ .  (16)

2. $H\left(\epsilon ,\beta \right)=1+\epsilon {\beta }^r$ , $r>0$ , in this way we obtain 

$D{P}_{\beta }^{\alpha }f\left(t\right):=\underset{\epsilon \to 0}{\text{lim}}\frac{\left(1+\epsilon {\beta }^r\right)f\left(t+\epsilon F\left(t,\alpha \right)\right)-f\left(t\right)}{\epsilon }$ .

Refer to our N-derivative of²⁴ we have:

$N{P}_2^{\alpha }f\left(t\right):=\underset{\epsilon \to 0}{\text{lim}}\frac{\left(1+\epsilon {\beta }^r\right)f\left(t+\epsilon e^{t^{-\alpha }}\right)-f\left(t\right)}{\epsilon }$ .  (17)

If $k\ne 1$ , as $e^x=1+x+\frac{x^2}{2!}+\mathrm{...}$ we can take (as a first possibility):

3. $H\left(\epsilon ,\beta \right)=E_{1,1}\left(\epsilon \beta \right)$  and so we have 

$D{E}_{\beta }^{\alpha }f\left(t\right):=\underset{\epsilon \to 0}{\text{lim}}\frac{E_{1,1}\left(\epsilon \beta \right)f\left(t+\epsilon F\left(t,\alpha \right)\right)-f\left(t\right)}{\epsilon }$ ,

and regarding our N-derivative of²⁴ it becomes:

$N{E}_{\beta }^{\alpha }f\left(t\right):=\underset{\epsilon \to 0}{\text{lim}}\frac{E_{1,1}\left(\epsilon \beta \right)f\left(t+\epsilon e^{t^{-\alpha }}\right)-f\left(t\right)}{\epsilon }$ .  (18)

From (15) we can easily obtain the following conclusions: 

1. Is a derivative local operator, that is, defined at a point.
2. They are derivative in the strict sense of the word.
3. It does not comply with Leibniz’s rule, so for (16) we have (the calculations are similar for (17) and (18)):

$N{L}_2^{\alpha }\left[f\left(t\right)g\left(t\right)\right]=\left(N_2^{\alpha }f\left(t\right)\right)g\left(t\right)+f\left(t\right)\left(N_F^{\alpha }g\left(t\right)\right)$ ,

Also for (16) we have (again the calculations for (17) and (18) are very similar):

4. If $\alpha =0$ , $\beta =1$ then  $N_2^{\alpha }f\left(t\right)=N_F^{0}f\left(t\right)+f\left(t\right)=\left(1+e\right)f\left(t\right)$ .
5. If $\alpha =1$ , $\beta =0$ then 

$N_2^{1}f\left(t\right)=N_{e^{t^{-1}}}^{1}f\left(t\right)=\underset{\epsilon \to 0}{\text{lim}}\frac{f\left(t+\epsilon e^{t^{-1}}\right)-f\left(t\right)}{\epsilon }=e^{t^{-1}}\left[\underset{\epsilon \to 0}{\text{lim}}\frac{f\left(t+\epsilon e^{t^{-1}}\right)-f\left(t\right)}{\epsilon e^{t^{-1}}}\right]=e^{t^{-1}}{f}^{\prime }\left(t\right)$

if f is derivable.

6. If the limit exists in (18) then we have

$N{L}_{\beta }^{\alpha }f\left(t\right)=N_F^{\alpha }f\left(t\right)+\beta f^{\prime }\left(t\right)$ .  (19)

7. Unfortunately, “we lose" the Chain Rule that was valid for our N-derivative (see²⁴), so for $N{L}_{\beta }^{\alpha }$ we obtain:

$N{L}_{\beta }^{\alpha }\left[f\left(g\left(t\right)\right)\right]=N_F^{\alpha }f\left(g\left(t\right)\right)+\beta f\left(g\left(t\right)\right)$ .

If $g\left(t\right)=t$ , the above expression is a generalization of proportional derivative of.⁵⁵

8. From (19) we derive that

$\underset{t\to \infty }{\text{lim}}N{L}_{\beta }^{\alpha }f\left(t\right)=\underset{t\to \infty }{\text{lim}}{N}_F^{\alpha }f\left(t\right)+\underset{t\to \infty }{\text{lim}}\beta f^{\prime }\left(t\right)=f^{\prime }\left(t\right)+\beta f\left(\infty \right)$ .

Where we can draw the following: if the term $\beta f\left(\infty \right)$ exists, then the derivative $N_{\beta }^{\alpha }f\left(t\right)$ is only a "translation" of the derivative of the function when $t\to \infty$ , so it does not affect the qualitative behavior of the ordinary derivative, this is of vital importance in the study of asymptotics properties of solutions of fractional differential equations with $N{L}_{\beta }^{\alpha }$ . Unfortunately, the non-existence of the limit of the function to infinity makes the qualitative study of these fractional differential equations impossible.

9. Let’s go back to the equation (15), it is clear that the function $H\left(\epsilon ,\beta \right)$ can be generalized although that would complicate the calculations extraordinarily. Of course this does not close the discussion on what terms can be “added" to the increased function that give local fractional derivatives that violate the Leibniz Rule, which would maintain the locality, as a historical inheritance of the derivative, and would default Leibniz’s Rule, as a “necessary" condition so that there is a fractional derivative.

In this paper, we have presented a sketch of the latest developments obtained in the Non-Integer Order Calculus. Of course, they are not all, for example in⁵⁶ a multi-index derivative is presented that generalizes the previous definitions and includes as a particular case the derivative presented in.⁵⁷

All of the above shows that this topic is a fruitful field and has not finished giving us good results.

None.

None.

1. Podlybny. Fractional differential equations, London, Acad. Press, 1999.
2. Kilbas AA, Srivastava HM, Trujillo JJ. Theory and application on fractional differential equations. North Holland Math Stud. 2006;204:7–10.
3. Lakshmikantham V, Leela S, Devi JV. Theory of fractional dynamic systems. Cambridge Scientific Publ. 2009.
4. Atangana A. Derivative with a New Parameter Theory, Methods and Applications. Academic Press. 2016.
5. S Umarov, S Steinberg. Variable order differential equations with piecewise constant order–function and diffusion with changing modes. Z Anal Anwend. 2009;28(4):431–450.
6. Atangana A. Extension of rate of change concept: From local to nonlocal operators with applications. Results in Physics. 2020;19:103515.
7. Capelas de Oliveira E, Tenreiro Machado JA. A Review of Definitions for Fractional Derivatives and Integral. Mathematical Problems in Engineering. 2014;2014(238459):1–6.
8. Kolwankar KM. Local fractional calculus: a review. Math Phys. 2013.
9. Vivas–Cortez, Paulo M Guzmán, Luciano M Lugo, et al. Fractional Calculus: Historical Notes. Revista Matua. 2021;8(2);1–13.
10. Kolwankar KM, Gangal AD. Fractional differentiability of nowhere differentiable functions and dimensions. Chaos. 1996;6(4):505–513.
11. Kolwankar KM, Gangal AD. Holder exponents of irregular signals and local fractional derivatives. Pramana J Phys. 1997;48(1):49–68.
12. Kolwankar KM, Gangal AD. Local fractional Fokker–Planck equation. Phys Rev Lett. 1998;80:214.
13. Parvate A D Gangal. Calculus on fractal subsets of real line – I: formulation. Fractals. 2009;17(1):53–81.
14. Yang X J. Local Fractional Integral Transforms. Progress in Nonlinear Science. 2011:1–225.
15. Yang X J. Local Fractional Functional Analysis and Its Applications. Hong Kong: Asian Academic publisher Limited, China. 2011.
16. Adda FB, Cresson J. About Non–differentiable Functions. Journal of Mathematical Analysis and Applications. 2001;263(2):721–737.
17. Chen W. Time–space fabric underying anomalous diffusion. Chaos, Solitons and Fractals. 2006;28(4):923–929.
18. He JH. A new fractal derivation. Thermal Science. 2011;15(1):145–147.
19. Gao F, Yang X, Kang Z. Local fractional Newton’s method derived from modified local fractional calculus. 2009 International Joint Conference on Computational Sciences and Optimization. 2009:228–232.
20. Yang X J. Transport Equations in Fractal Porous Media within Fractional Complex Transform Method, Proceedings. Romanian Academy, Series A. 2013;14(4):287–292.
21. Yang X J. Local Fractional Integral Transforms and Their Applications, Academic Press, New York, USA. 2015.
22. Khalil R, Al Horani M, Yousef A, et al. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65–70.
23. Abdeljawad T. On conformable fractional calculus. Journal of Computational and Applied Mathematics. 2015;279:57–66.
24. Bohner M, Kashuri A, Mohammed PO, et al. Hermite–Hadamard–type inequalities for conformable integrals. Hacet J Math Stat. 2022;51(3):775–786.
25. Guzmán PM, Langton G, Lugo LM, et al. A new definition of a fractional derivative of local type. J Math Anal. 2018;9(2):88–98.
26. Valdés JEN, Guzmán PE, Lugo LM. Some new results on non–conformable fractional calculus. Adv Dynamical Sys Appl. 2018;13(2):167–175.
27. Abreu–Blaya R, Fleitas A, Nápoles Valdés JE, et al. On the conformable fractional logistic models. Math Meth Appl Sci. 2020;43(7):4156–4167.
28. Fleitas A, Gómez–Aguilar JF, Nápoles Valdés JE, et al. Analysis of the local Drude model involving the generalized fractional derivative. Optics. 2019;193:163008.
29. Fleitas A, Méndez–Bermúdez JA, Nápoles Valdés JE, et al. On fractional Liénard–type systems. Revista Mexicana de Fisica. 2019;65(6):618–625.
30. Guzmán PM, Nápoles Valdés JE. A note on the oscillatory character of some non–conformable generalized lienard system. Advanced Mathematical Models & Applications. 2019;4(2):127–133.
31. Guzmán PM, Lugo LM, Nápoles JE. On the stability of solutions of fractional non conformable differential equations. Stud Univ Babes–Bolyai Math. 2020;65(4):495–502.
32. Guzmán PM, Lugo LM, Nápoles JE. A note on the qualitative behavior of some nonlinear local improper conformable differential equations. J Frac Calc & Nonlinear Sys. 2020;1(1):13–20.
33. Martínez F, Mohammed FPO, Valdés JEN. Non–conformable fractional Laplace transform. Kragujevac J Math. 2022;46(3):341–354.
34. Martínez F, Valdés JEN. Towards a non–conformable fractional calculus of n–variables. J Math Appl. 2020;43:87–98.
35. JE Nápoles, JM Sigarreta. New Hermite–Hadamard type inequalities for non–conformable integral operators. Symmetry. 2019;11(9):1108.
36. JE Nápoles, C Tunc. On the Boundedness and Oscillation of Non–Conformable Lienard Equation. Journal of Fractional Calculus and Applications. 2020;11(2):92–101.
37. Valdés JEN, Guzmán PM, Lugo LM, et al. The local generalized derivative and Mittag Leffler function. Sigma J Eng Nat Sci. 2020;38(2):1007–1017.
38. Fleitas A, Nápoles JE, Rodríguez JM, et al. Note on the generalized conformable derivative. Revista de la UMA. 2021;62(2):443–457.
39. Zhao D, Luo M. General conformable fractional derivative and its physical interpretation. Calcolo 54. 2017;903–917.
40. Chatzarakis GE, Nápoles Valdés JE. Continuability of Lienard’s Type System with Generalized Local Derivative. Discontinuity, Nonlinearity, and Complexity. 2023;12(1):1–11.
41. Guzmán PM, Kórus P, Nápoles Valdés JE. Generalized Integral Inequalities of Chebyshev Type. Fractal Fract. 2020;4(2):10.
42. Guzmán PM, Nápoles JE, Gasimov Y. Integral inequalities within the framework of generalized fractional integrals. Fractional Differential Calculus. 2021;11(1):69–84.
43. Hernández–Gómez JC, Nápoles Valdés JE, Rosario–Cayetano O, et al. Energy–Level Spacing Distribution of Quantum Systems via Conformable Operators. Discontinuity, Nonlinearity, and Complexity. 2022;11(2):325–335.
44. Kórus P, Lugo LM, Valdes JEN. Integral inequalities in a generalized context. Studia Scientiarum Mathematicarum Hungarica. 2020;57(3):312–320.
45. Lugo JM, Nápoles JEV, Vivas–Cortez M. On the oscillatory behaviour of some forced nonlinear generalized differential equation. Investigacion Operacional. 2021;42(2)267–278.
46. Mehmood S, Shahzad M, Batool K, et al. Some integral inequalities in the framework of conformable fractional integral. J Prime Res Math. 2021;17(2):159–167.
47. Nápoles JE. Oscillatory criteria for some non–conformable differential equation with damping. Interdis J Discontinuity, Nonlinearity Complexity. 2021;10(3):461–469.
48. Valdés JEN, Gasimov YS, Aliyeva AR. On the oscillatory behavior of some generalized differential equation. Punjab Univ J Math. 2021;53(1):71–82.
49. Valdés JEN, Jose SA. A note on the boundedness of solutions of generalized functional differential equations. Appl Math E–Notes. 2022;22:265–272.
50. Nápoles JE, Quevedo MN. On the oscillatory nature of some generalized emden–fowler equation. Punjab Univ J Math. 2020;52(6):97–106.
51. Nápoles JE, Quevedo MN, Plata ARG. On the asymptotic behavior of a generalized nonlinear equation. Sigma J Eng Nat Sci. 2020;38(4):2109–2121.
52. Vivas–Cortez, P Kórus, JE Nápoles Valdés. Some generalized Hermite–Hadamard–Fejér inequality for convex functions. Advances in Difference Equations. 2021:199.
53. Vivas–Cortez J, E Nápoles Valdés, LM Lugo. On a Generalized Laplace Transform. Appl Math Inf Sci. 2021;15(5):667–675.
54. Mingarelli B. On generalized and fractional derivatives and their applications to classical mechanics. Math Phys. 2018.
55. Zulfeqarr F, Ujlayan A, Ahuja A. A new fractional derivative and its fractional integral with some applications. 2017.
56. Anderson DR, Ulness DJ. Newly defined conformable derivatives. Adv Dyn Sys App. 2015;10(2):109–137.
57. Vivas–Cortez M, Lugo LM, Juan E Nápoles Valdés, et al. A Multi–Index Generalized Derivative Some Introductory Notes. Appl Math Inf Sci. 2022;16(6):883–890.
58. E Reyes–Luis, G Fernández–Anaya, J Chávez–Carlos, et al. A two–index generalization of conformable operators with potential applications in engineering and physics. Revista Mexicana de Física. 2021;67(3):429–442.
59. Katugampola UN. A new fractional derivative with classical properties. J Amer Math Soc. 2014;1:1–8.