On more general forms of proportional fractional operators

In this article, we propose new proportional fractional operators generated from local proportional derivatives of a function with respect to another function. We present some properties of these fractional operators which can be also called proportional fractional operators of a function with respect to another function or proportional fractional operators with dependence on a kernel function.


Introduction
The fractional calculus, which is engaged in integral and di erential operators of arbitrary orders, is as old as the conceptional calculus that deals with integrals and derivatives of non-negative integer orders. Since not all of the real phenomena can be modeled using the operators in the traditional calculus, researchers searched for generalizations of these operators. It turned out that the fractional operators are excellent tools to use in modeling long-memory processes and many phenomena that appear in physics, chemistry, electricity, mechanics and many other disciplines. Here, we invite the readers to read [1][2][3][4][5][6][7][8][9][10] and the reference cited in these books. However, targeting the best understanding more accurate modeling real world problems, researchers were in need of other types of fractional operators that were con ned to Riemann-Liouville fractional operators. In the literature, one can nd many works that propose new fractional operators. We mention [11][12][13][14][15][16]. Nonetheless, the fractional integrals and derivatives which were proposed in these works were just particular cases of what so called fractional integrals/derivatives withe dependence on a kernel function [2,5,17]. There are other types of fractional operators which were suggested in the literature.
On the other hand, due to the singularities found in the traditional fractional operators which are thought to make some di culties in the modeling process, some researches recently proposed new types of nonsingular fractional operators. Some of these operators contain exponential kernels and some of them involve the Mittag-Le er functions. For such types of fractional operators we refer to [18][19][20][21][22][23][24][25][26][27].
All the fractional operators considered in the references in the rst and the second paragraphs are non-local. However, there are many local operators found in the literature that allow di erentiation to a non-integer order and these are called local fractional operators. In [28], the authors presented what they called conformable (fractional) derivative. The author in [29] proposed other basic concepts related to the conformable derivatives. We would like to mention that the fractional operators proposed in [12,13] are the non-local fractional version of the local operators suggested in [28]. In addition, the non-local fractional version of the ones in [29] can be seen in [16].
It is customary that any derivative of order 0 when performed to a function should give the function itself. This essential property is dispossessed by the conformable derivatives. Notwithstanding, in [30,31], for skae of overcoming this obstacle, the authors proposed a new de nition of the conformal derivative that gives the function itself when the order of the local derivative approaches 0. In addition to this, the non-local fractional operators that emerge from iterating the above-mentioned derivative were held forth in [32].
In this article, we extend the work done in [32] to introduce a new fractional operators relying on the proportional derivatives of a function with respect to another function which can be de ned in parallel with the de nitions discussed in [30]. The kernel obtained in the fractional operators which will be proposed contains an exponential function and is function dependent. The semi-group properties will be discussed.
The article is organized as follows: Section 2 presents some essential de nitions for fractional derivatives and integrals. In Section 3 we present the general forms of the fractional proportional integrals and derivatives. In section 4, we present the general form of Caputo fractional proportional derivatives. In the end, we conclude our results.

Preliminaries
In this section, we present some principal de nitions of fractional operators. We rst present the traditional fractional operators and then the fractional proportional operators.

. The conventional fractional operators and their general forms
For ω ∈ C, Re(ω) > , the forward (left) ωth order Riemann-Liouville fractional integral is de ned by (2.1) The backward (right) ωth Riemann-Liouville fractional integral reads The forward ωth order Riemann-Liouville fractional derivative, whre Re(ω) ≥ is given as 3) The backward ωth order Riemann-Liouville fractional derivative, where Re(ω) ≥ reads The forward Caputo fractional derivative has the following form The backward Caputo fractional derivative reads The generalized forward and backward fractional integrals Katugampola setings [12] are given respectively as and The generalized forward and backward fractional derivatives in the sense of Katugampola [13] are de ned respectively as where σ > and γ = x −σ d dx . The Caputo modi cation of the forward and backward generalized fractional derivatives are proposed in [14] in the following forms respectively and For ω ∈ C, Re(ω) > the forward Riemann-Liouville fractional integral of order ω of a function f with respect to a continuously di erentiable and increasing function ν has the following form [2,5] For ω ∈ C, Re(ω) > the backward Riemann-Liouville fractional integral of order ω of f with respect to a continuously di erentiable and increasing function ν has the following form [2,5] For ω ∈ C, Re(ω) ≥ , the generalized forward and backward Riemann-Liouville fractional derivatives of order ω of f with respect to a continuously di erentiable and increasing function ν have respectively the forms [2,6] and  [2,5]. While if one considers ν(x) = x σ σ , the fractional operators in the settings of Katugampola [12,13] are derived.
In forward and backward generalized Caputo derivatives of a function with respect to another function are presented respectively as [

. The proportional derivatives and their fractional integrals and derivatives
In [28], the authors introduced The conformable derivative. More properties and a modi ed type of this derivative were explored in [29]. [ and µ (σ, t) ≠ , σ ∈ [ , ), µ (σ, t) ≠ , σ ∈ ( , ]. Then, the modi ed conformable di erential operator of order σ is de ned by For details about such derivatives we refer to [30,31]. As a special case, we shall consider the simplest case and restrict our work to the case when µ (σ, t) = −σ and µ (σ, t) = σ. Therefore, (2.19) becomes (2.20) It is obvious that the derivative (2.20) is generalizes the conformable derivative which does not yieldo the original function as σ approaches to . The associated fractional proportional integrals are de ned as follows.

De nition 2.2. [32]
For σ > and ω ∈ C, Re(ω) > , the forward fractional proportional integral of f reads and the backward one reads

De nition 2.3. [32]
For σ > and ω ∈ C, Re(ω) ≥ , the forward fractional proportional derivative is de ned as The backward proportional fractional derivative is de ned by [32] ( Lastly, the left and right fractional proportional derivatives in the Caputo settings respectively read [32] (

The fractional proportional derivative of a function with respect to another function
and µ (σ, t) ≠ , σ ∈ [ , ), µ (σ, t) ≠ , σ ∈ ( , ]. Let also ν(t) be a strictly increasing continuous function. Then, the proportional di erential operator of order σ of f with respect to g is de ned by We shall restrict ourselves to the case when µ (σ, t) = − σ and µ (σ, t) = σ. Therefore, (3.1) becomes The corresponding integral of (3.2) where we accept that a I ,σ f (t) = f (t).
To generalize a more general class of fractional integral based on the proportional derivative, we use induction and changing the order of integrals to show that , ω ∈ C, Re(ω) > , we de ne the left fractional integral of f with respect to g by The right fractional proportional integral ending at b can be de ned by
Proof. The proofs of relations (a) and (b) are very easy to handle. We will prove (c) while the proof of (d) is analogous. By the de nition of the left proportional fractional derivative and relation (a), we have Here, we have used the fact that Below we present the semi-group property for the general fractional proportional integrals of a function with respect to another function.
Proof. We only prove the rst relation. The proof of the second relation is similar. We have

Conclusions
We have used the proportional derivatives of a function with respect to another function to obtain left and right generalized type fractional integrals and derivatives involving two parameters ω and σ and depending on a kernel function. The Riemann-Liouville and Caputo fractional derivatives in the classical fractional calculus can obtained as σ tends to and by choosing ν(t) = t. The integrals have the semi-group property and together with their corresponding derivatives have exponential functions as part of their kernels. It should be noted that other properties of these new operators can be obtained by using the Laplace transform proposed in [17]. Moreover, for a speci c choice of ν, the proportional fractional operators in the settings of Hadamard and Katugampola can be obtained.