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

Muhammad Samraiz,Sajid Iqbal,Artion Kashuri,Muhammad Umer,Nabil Mlaiki,Thabet Abdeljawad

doi:10.3390/fractalfract5030118

Abstract

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.

Highlights

Fractional calculus history dates back to the 17th century, when the derivative of order α = 1/2 was defined by Leibnitz in 1695
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

Summary

Introduction

The (k, s)-Riemann-Liouville fractional integral (RLFI) [21] is given in the following definition. We define the weighted (k, s)-Riemann Liouville fractional operators and discuss some of their properties. The (k, s)-Riemann Liouville fractional derivative is defined as follows: Let φ be a continuous function on [0, ∞) and s ∈ R\{−1}. If ω (ξ ) = 1 is chosen, we obtain the (k, s)-Riemann-Liouville fractional derivative [22]. Let s = 0 and ω (ξ ) = 1, where it gives the k-Riemann-Liouville fractional derivative [26]. We present the space where the weighted (k, s)-Riemann-Liouville fractional integrals are bounded. If we choose s = 0, k = 1 and ω (ξ ) = 1 in Theorem 5 and Corollary 5, we obtain results for Riemann Liouville.

The Weighted Laplace Transform of the Weighted Fractional Operators

Fractional Kinetic Differ-Integral Equation

Conclusions and Discussion