Type of Leibniz Rule on Riemann-Liouville Variable-Order Fractional Integral and Derivative Operator

TL;DR

This paper develops multiple types of Leibniz rule formulas, including product, quotient, and chain rules, for Riemann-Liouville variable-order fractional operators, expanding the theoretical framework of fractional calculus.

Contribution

It introduces four types each of product, quotient, and chain rule formulas for variable-order fractional derivatives and integrals, providing a comprehensive set of rules for this operator class.

Findings

Four types of product rule formulas established.

Four types of quotient rule formulas developed.

Four types of chain rule formulas formulated.

Abstract

In this paper, types of Leibniz Rule for Riemann-Liouville Variable-Order fractional integral and derivative Operator is developed. The product rule, quotient rule, and chain rule formulas for both integral and differential operators are established. In particular, there are four types of product rule formulas: Product rule type-I, Product rule type-II, Product rule type-III, and Product rule type-Iv. Quotient rule type-I, quotient rule type-II, quotient rule type-III, and quotient rule type-Iv formulas developed from product rule types. There are four types of chain rule formulas: chain rule type-I, chain rule type-II, chain rule type-III, and chain rule type-Iv.