WEIGHTED FRACTIONAL PROPORTIONAL OPERATORS REGARDING A FUNCTION AND THEIR HILFER UNIFICATION

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 ð-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.