On Weighted ( k , s ) -Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation

: In this article, we establish the weighted ( k , s ) -Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms.


Introduction
Fractional calculus history dates back to the 17th century, when the derivative of order α = 1/2 was defined by Leibnitz in 1695. Fractional calculus has gained broad significance in the last few decades due to its applications in various fields of science and engineering. The Tautocrone problem can be solved using fractional calculus, as shown by Abel [1]. It also has applications in group theory, field theory, polymers, continuum mechanics, wave theory, quantum mechanics, biophysics, spectroscopy, Lie theory, and in several other fields [2][3][4][5][6]. Despite the fact that this calculus is ancient, it has gained attention over the last few decades because of the interesting results derived when this calculus is applied to the models of some real-world problems [7][8][9][10][11][12][13][14]. The fact that there are various fractional operators is what makes fractional calculus special. Thus, any scientist working on modeling real global phenomena can choose the operator that best suits the model.
The Riemann-Liouville, Grünwald-Letnikov, and Caputo and Hadamard definitions [7,15,16] are some of the most well-known definitions of fractional operators, such that their formulations include single-kernel integrals, and they are used to explore and analyze memory effect problems, for example [17]. The fractional derivatives are represented by the fractional integrals [7,10,15,18] in fractional calculus. There are several varieties of fractional integrals, of which two have been studied extensively for their applications. The first one is the Riemann-Liouville fractional integral defined for parameter β ∈ R + by well-defined for n ∈ N. The second is Hadamard's fractional integral, which is defined by Hadamard [19] (J β a ϕ) ( and is derived by the following integral: We start by recalling some related results and notions. Definition 1 ([20]). The integral form of the k-gamma function is defined by Definition 2. Let (α), (β) > 0 and k > 0, where we have the following k-beta function Note that the relation between Γ k and B k functions is given by The (k, s)-Riemann-Liouville fractional integral (RLFI) [21] is given in the following definition.
Theorem 1 ([23]). The generalized weighted Laplace transform of D n ω ϕ exists and is given by

Weighted (k, s)-Riemann Liouville Fractional Operators
In the present section, we define the weighted (k, s)-Riemann Liouville fractional operators and discuss some of their properties.

Remark 1.
It should be noted that this integral operator covers many fractional integral operators.
The following modification of Definition 4 is required to prove the claimed results.
It can also be written as Remark 2. It is worth mentioning that many other derivative operators can be represented as special cases of (6).
Next, we present the space where the weighted (k, s)-Riemann-Liouville fractional integrals are bounded.
Substituting ξ s+1 = v and t s+1 = u on the right side of (7), we obtain By using Minkowski's inequality, we have Applying Hölder's inequality, we obtain Hence, we obtain the desired result.

Proof. Consider
By substituting z = y s+1 −t s+1 ξ s+1 −t s+1 on the right side of (8), we obtain The inverse property is proved.
Proof. By using Definition 9, we have By using Theorem 3, we have , which is the required result.
where Γ k denotes the k-Gamma function.
Proof. Since ϕ and ψ are synchronous on [0, ∞), for all ξ, y ≥ 0, we have Both sides of (14) are multiplied by α k −1 ω(ξ)ξ s and integrating w.r.t ξ over (a,t), we obtain which gives Both sides of (15) are multiplied by α k −1 ω(y)y s and integrating w.r.t y over (a,t), we obtain This can be written as On simplification, we obtain which can be written as This completes the proof of (12).
Both sides of (16) are multiplied by β k −1 ω(y)y s and integrating w.r.t y over (a,t), we obtain s k J α a + ,ω ϕψ(t) The proof of (13) is done.
Proof. Since the function ϕ and ψ are synchronous on [0, ∞), h ≥ 0, for all α, β > 0, we have This gives Both sides of (18) are multiplied by β k −1 ω(ξ)ξ s and integrating w.r.t ξ over (a,t), we obtain Proof. By using Definition 7 and the Dirichlet's formula, we have Hence, we obtained the desired result.

The Weighted Laplace Transform of the Weighted Fractional Operators
In this section, we apply the weighted laplace transformation to the new fractional operators. For this purpose we need to substitute ψ(t) = t s+1 on the right side of (3), where we have which holds for all values of u.
Proof. By using (26), we have By substituting t = (ξ s+1 − a s+1 ) on the right side of (27), we obtain which gives the required result.
Proof. By using Definitions 6 and 7 and Proposition 1, we have This proves the claimed result.
Theorem 12. The Laplace transform of the weighted (k, s)-Riemann Liouville derivative is given by Proof. By using Definition 9, Theorems 1 and 11, we obtain which gives the required series solution.

Fractional Kinetic Differ-Integral Equation
The fractional differential equations are significant in the field of applied science and have gained interest in dynamic systems, physics, and engineering. In the previous decade, the fractional kinetic equation has gained interest due to the discovery of its relationship with the CTRW theory [28]. The kinetic equations are essential in natural sciences and mathematical physics that explain the continuation of motion of the material. The generalized weighted fractional kinetic equation and its solution related to novel operators are discussed in this section. Consider the fractional kinetic equation given by with initial condition where α ≥ 0, a, b ∈ R(a = 0), k > 0, n = [ α k ] = 1.
Theorem 13. The solution of (30) with initial condition (31) is Proof. Applying the modified weighted Laplace transform on both side of (30), we obtain Using Theorems 11 and 12, we obtain Applying inverse Laplace transform, we obtain The proof of the result is completed.

Conclusions and Discussion
Fractional calculus is currently one of the most widely debated topics. In the present article, we introduced the weighted versions of the (k, s)-RLF operators. We then investigated and examined their properties and found the weighted Laplace transform of the new operators. Significantly, these operators reduce to notable fractional operators in the literature. Other fractional operators, such as the Riemann-Liouville fractional operators and Hadamard fractional operators, show up as special cases of these weighted fractional operators with specific choices of weighted functions and operator functions. We have developed the Chebyshev inequalities by involving the introduced fractional integral operator. We developed a fractional kinetic equation and the weighted Laplace transform used to find the solution of the said model. The presented results motivate scientists to stimulate more work in such directions. Funding: There is no funding available for this work.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
No data were used to support this study.