1.Introduction

Derivatives of non-integer order have been discussed for a long time. In 1695 L’Hôpital asked Leibniz what meaning could be ascribed to Dnf if n were a fraction. Since that time, fractional calculus has drawn the attention of many mathematicians. Most of them have used an integral form to define the fractional derivative; the two most popular ones are the Riemann-Liouville definition and the Caputo definition. However, these definitions did not inherit most of the properties of the classic integer-order derivative, such as the product rule or the chain rule. In 2014 Khalil et al. ¹ introduced the definition of the conformable fractional derivative, analogous to the limit definition of the standard integer-order derivative. This definition was later generalized by Katugampola ², whose work was later addressed by Anderson et al. ³, who used Katugampola’s results to explore its properties and its potential application in quantum mechanics, see also ^4-8. In 2017, Al-Refai and Abdeljawad ⁹ suggested a generalization of the well-known Sturm-Liouville eigenvalue problem using the conformable derivative. In 2018 Zhao et al. ¹⁰ introduced a set of Maxwell’s equations using the Generalized Conformable Fractional Derivative (GCFD), which obeys the classical properties of the integer-order derivative. Several works have recently appeared on conformal derivatives, e.g., ^14-19. Clearly showing that it is a field of great growth, despite its recent creation. Some applications of these conformal derivatives to optical solitons solutions are in ^20,21.

We generalize the conformable derivative introduced by Khalil et al. ¹ by exploring the properties of a fractional derivative operator derived from the GCFD derivative. Unlike the rest of the fractional derivatives, the order of our operator is determined by two fractional indexes and two positive functions, which can be arranged to recover the integer order derivative. The two fractional indexes and the two functions give the derivative greater freedom and more complex dynamics than single-index and single-function derivatives; furthermore, being a conformable derivative, it is a local derivative just like in the integer-order case. In this way, our proposed GCFD is similar to the Gateaux derivative ¹⁰ but definitively is not equal. We present a differential operator in which the order of the derivative now depends on the two fractional indexes α and β, preserving almost all the properties of integer-order derivatives, as it is often the case with conformable local derivatives. We construct operators with potential application in quantum mechanics, generalizing some results obtained by Anderson and Ulness ³. We generalize some well-known results, such as the Euler-Lagrange method ¹¹ and the Sturm-Liouville eigenvalue problem ¹² by replacing the integer-order derivative operators with our operator, following a similar procedure as Abdeljawad et al. ⁹ and we show some examples.

In the first section, we present the definition of the operator, its main properties, such as the product rule and the chain rule, as well as some examples. In the next section, we further explore its properties, we introduce an inverse operator, and then we explore its commutator and anti-commutator properties, as well as some results with applications in quantum mechanics. In the last section, we generalize some results obtained in the second section, namely the Sturm-Liouville operator. We consider the Sturm-Liouville eigenvalue problem using the differential operator introduced in the first section. We also generalize some results, such as the Lagrange Identity and the Euler-Lagrange equation.

2.Preliminaries

Let ψ(u,α,β) be a fractional conformable function and α,β∈ (0,1]. The “αβ derivative” of f: R+→R at x is defined as:

The restrictions of ψ(u,α,β) are:

ψ(x,1,1)=1x ∈ R+ψ(x,α,β)≠0x ∈ R+ψ(.,α1,β)≠ψ(.,α2,β)α1≠α2ψ(.,α,β1)≠ψ(.,α,β2)β1≠β2

where αi and βi ∈ (0,1], for i =1, 2, 3…. Let f: [0, ∞ ] →,R and x > 0 also let h(x) and k(x) be continuous functions, where h(x)>0 and k(x)>0. Then the “α,βhkl conformable fractional derivative” of f of order α, β is defined by:

where

for x > 0, α,β ∈ (0,1]. If the limit exists, we say that f (x) is γ-differentiable.

Recently M.N. Alam and Xin Li ¹³ solved the complex fractional Schrödinger equation using the conformable derivative introduced by Khalil et al. ¹; our fractional operator generalizes the operator introduced by Khalil.

If we set h(x)=1, β = 1, k(x)=x and l =1 then we recover the conformable fractional derivative definition from Khalil et al.:

If we set h(x) = x, β = 1, k(x)=xαα-1 and l→∞ then we recover the fractional derivative of order α definition from Katugampola:

3.Properties of the differential operator

Let α ∈ (0,1] and f,g be γ -differentiable at a point x > 0. Using (2) and (2), it can be easily proved that:

4.The αβhkl conformable fractional derivate and its relationship with second-order linear defferential equations

We now further explore the properties of our differential operator (2) and develop some new operators from this one, such as the inverse operator, the commutator and the anti-commutator as well as a self-adjoint variant of the anti-commutator. We will later use some of this results when solving the Sturm-Liouville eigenvalue problem, such as the integration by parts formula.