A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law

In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann–Liouville involving a new modified version of a generalized Mittag–Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann–Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.


Introduction
Recently, fractional calculus has developed as an exciting and widely appreciated subject of research.There are various research papers referring to fractional differential equations and inclusions with a variety of boundary conditions published in this topic.These researches include the theoretical evolvement, numerical strategies, as well as comprehensive applications in the mathematical modeling of biological, physical, and engineering phenomena, we refer readers to the works [1][2][3][4][5][6][7][8][9][10][11][12] and references cited therein.
In fractional calculus, there appeared two types of nonlocal operators, the first type is with a singular kernel such as Caputo [3], Riemann-Liouville [3], Katugampola [13], Hadamard [3], and Hilfer [2] operators.Also, these operators were extended to weighted fractional operators with respect to another function [14][15][16][17].In fact, the singularity of kernels caused problems sometimes in numerical analysis.For this reason, the second type of nonlocal operators appeared that created a new fractional operator with nonsingular kernel by Caputo and Fabrizio [18] in 2015.Then, Atangana and Baleanu (AB) [19] introduced a new nonsingular fractional operator via a function of Mittag-Leffler.These two operators attracted the attention of many researchers to study various problems with applications [20].Afterwards, Fernandez and Baleanu [21] generalized the AB operator to differointegration with respect to another function, and recently Abdeljawad et al. [22] extended this to higher fractional orders.In 2018, Abdeljawad and Baleanu [23] derived a new AB operator with the generalized Mittag-Leffler kernels of three parameters.In 2019, Al-Refai and Jarrah [24] defined the weighted Caputo-Fabrizio-Caputo (CFC) fractional operator respect to another function.In 2020, the weighted Atangana-Baleanu-Caputo (ABC) fractional derivative was presented by Al-Refai [25].Also, in 2020 Hattaf [26], proposed a new fractional operator of a nonsingular kernel in the Caputo and Riemann-Liouville sense that is generalized to some definitions existing in the literature.
Motivated by the above various operators of nonsingular kernels, we aim in this paper to present new generalized weighted fractional derivatives and integrals with respect to another function, which cover all existing forms of nonlocal, nonsingular fractional derivatives and integrals.For this, in Section 2 some definitions and basics are presented.Section 3 is concerned with new generalized weighted fractional derivatives with respect to another function in the sense of Caputo and Riemann-Liouville, as well as their corresponding fractional integrals.Indeed, this section proves many propositions and lemmas related to our fractional derivatives and integrals.

Background materials
In this section, we present some background materials, which support us in order to reach the desired results.

Main results
In this section, we will introduce new weighted generalized fractional derivatives and integrals with respect to another function.

New fractional derivatives
First, we will define a generalized weighted fractional derivative in the Riemann-Liouville sense with respect to another function.
The new αth left-sided generalized weighted fractional derivative in the Caputo sense of a function g with respect to another function φ(u) is given by where (α) is the normalization function with (0 β,μ is the modified version of the generalized Mittag-Leffler function defined in (2.4).Remark 3.4 Note that for any specific values for the parameters α, β, γ , μ, σ , φ, ω, we obtain new and known definitions in the literature as follows: 1.If μ = σ = 1 and φ(u) = u in the new Def.3.3, we obtain the Hattaf-Caputo fractional derivative [26], which is given by and ω(u) = 1 in the new Def.3.3, we obtain the generalized ABC fractional derivative with the generalized Mittag-Leffler function [23], which is given by , and φ(u) = u in the new Def.3.3, we obtain the weighted ABC fractional derivative [25], which is given by and ω(u) = 1 in the new Def.3.3, we obtain the ABC fractional derivative of g with respect to another function φ(u) [21], which is given by and ω(u) = 1 in the new Def.3.3, we obtain the ABC fractional derivative [19], which is given by , we obtain the weighted CFC fractional derivative of g with respect to another function φ(u) [24], which is given by and ω(u) = 1 in the new Def.3.3, we obtain the CFC fractional derivative of g with respect to another function φ(u) [24], which is given by and ω(u) = 1 in the new Def.3.3, we obtain the CFC fractional derivative [18], which is given by respectively, where R J kγ +μ-1 ι,ω,φ and R J kγ +μ ι,ω,φ are defined as in (2.1).

, and the weight function
Proof By Lemma (2.2), part (iii), we obtain Proof In view of Propositions 3.6 and 3.7, we obtain By applying the inverse Laplace, the proof is finished.

The new corresponding fractional integral
In this subsection, we introduce a new generalized weighted fractional integral with respect to another function, and we discuss some properties relating to new fractional derivatives and integral definitions.
Proof In view of Proposition 3.6, for γ = β, we have Then, for σ = 1, and applying the Laplace transform on both sides of ( R D α,β,β,μ ι,1,ω,φ g)(u) = f (u) and by using (3.5), we find where, F(s) = L ω,φ {f (u)}(s) and G(s) = L ω,φ {g(u)}(s), which yields that By taking the inverse Laplace transform and using Lemma 2.2, we obtain Now, for σ = 2, we have Thus, it implies that Due to the inverse Laplace transform and Lemma 2.2, we obtain Hence, by the same manner upto σ = n, one has which completes the proof. where is defined in (2.1).

Conclusions
Fractional operators with Mittag-Leffler kernels, whose integer order as a limiting case by means of delta Dirac functions, have played an important role in developing the theory of fractional calculus.The corresponding fractional integrals of the ABC and ABR fractional operators with one parameter α consist of certain linear combination of the function itself and a Riemann-Liouville-type fractional integral that forces their linear fractional equations with constant coefficients to have a trivial solution.Adding a Mittag-Leffler function with three parameters inside the kernel of ABC enables us to obtain nontrivial solutions.In addition, the kernel may be singular in some cases.In this article, we have defined and investigated the weighted fractional operators, both derivatives and integrals, with respect to another function and with modified three-parameter Mittag-Leffler kernels.Such operators allow us to obtain operators with exponential kernels as special cases.
In addition, several new and existing operators of nonsingular and singular kernels were obtained as special cases.Further, we have studied many important properties related to our new operators, such as series versions involving Riemann-Liouville fractional integrals and weighted Laplace transform with respect to another function.

Figure 1
Figure 1 Plots of Example 3.15, for various values of α.
and g(u) is continuous on [ι, τ ].The new αth left-sided generalized weighted fractional derivative in the Riemann-Liouville sense of a function g with respect to another function φ(u) is given by