Weighted fractional calculus: a general class of operators

TL;DR

This paper introduces and studies a broad class of weighted fractional calculus operators, unifying various known types and enabling new solutions to differential equations through generalized transforms.

Contribution

It formalizes the weighted fractional calculus class and extends it to functions, connecting it with classical operators and enabling generalized solution methods.

Findings

Contains tempered, Hadamard-type, Erdélyi–Kober operators as special cases

Relates these classes to classical Riemann–Liouville calculus via conjugation

Enables solving differential equations using modified Laplace transforms

Abstract

The operators of fractional calculus come in many different types, which can be categorised into general classes according to their nature and properties. We conduct a formal study of the class known as weighted fractional calculus and its extension to the larger class known as weighted fractional calculus with respect to functions. These classes contain tempered, Hadamard-type, and Erd\'elyi--Kober operators as special cases, and in general they can be related to the classical Riemann--Liouville fractional calculus via conjugation relations. Considering the corresponding modifications of the Laplace transform and convolution operations enables differential equations to be solved in the setting of these general classes of operators.