Weighted Fractional Proportional Operators Regarding a Function and Their Hilfer Unification

Abstract

In this paper, some new forms of fractional operators are proposed. These new forms are developed by using the proportional and the weighted derivative of a function regarding a function, known as weighted fractional proportional operators regarding another function. Additionally, the partial derivative-Hilfer version of the weighted proportional fractional derivatives, which is a concept that unifies the Riemann-Liouville and Caputo weighted proportional fractional derivatives, is propounded. Moreover, a number of fundamental properties of these operators and related important results are investigated. The Laplace transforms of the newly defined operators are found. Finally, we solve a particular type of differential equations involving the introduced derivatives in favor of the weighted Laplace transform.