The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only

TL;DR

This paper introduces a duality-based approach to fractional calculus, establishing integration by parts formulas involving only one type of fractional operator, and develops a new fractional calculus of variations with practical applications to dissipative systems.

Contribution

It presents a novel duality framework transforming left and right fractional operators, and introduces a new fractional calculus of variations using only one operator type.

Findings

New integration by parts formulas involving only left or right fractional operators

A fractional variational principle leading to equations of motion for dissipative systems

A practical Lagrangian relying solely on left fractional derivatives

Abstract

Through duality it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or right operators. The emergence of these novel fractional integration by parts formulas inspires the introduction of a new calculus of variations, where only one type of fractional derivative (left or right) is present. This applies to both the problem formulation and the corresponding necessary optimality conditions. As a practical application, we present a new Lagrangian that relies solely on left-hand side fractional derivatives. The fractional variational principle derived from this Lagrangian leads us to the equation of motion for a dissipative/damped system.