Weighted Fractional Calculus with respect to Functions

Abstract

This thesis explores the domain of fractional calculus, with a particular focus on generalized fractional calculus and weighted fractional calculus with respect to functions. The core research focuses on further exploring the existing theory of weighted fractional calculus with respect to functions. This theory seeks to prove important results for fractional differential equations of weighted Caputo fractional derivatives with respect to functions that have not yet been explored. By investigating the fundamental principles of these operators, weestablish mean value theorems, Taylor’s theorems, and integration by parts formulas. Our study extends to the Leibniz rule for weighted Riemann–Liouville derivatives with respect to functions. Additionally, we establish the existence and uniqueness theorems for a class of initial value problems involving weighted Caputo fractional derivatives with respect to functions in the Sobolev space.