Nonsymmetric and Symmetric Fractional Calculi on Arbitrary Nonempty Closed Sets

Abstract

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated, and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors.

This is a preprint of a paper whose final and definite form will be published in Mathematical Methods in the Applied Sciences, ISSN 0170-4214. Submitted 05/March/2014; revised 24/Feb/2015; accepted 25/Feb/2015. arXiv admin note: text overlap with arXiv:1405.2813

References in corpus (14)

• Discrete-Time Fractional Variational Problems
• Fractional $h$-difference equations arising from the calculus of variations
• Necessary Optimality Conditions for Fractional Difference Problems of the Calculus of Variations
• A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration
• Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform
• The diamond-alpha Riemann integral and mean value theorems on time scales
• Symmetric Differentiation on Time Scales
• The Diamond Integral on Time Scales
• A symmetric quantum calculus
• Hahn's Symmetric Quantum Variational Calculus
• Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales
• Fractional Calculus on Time Scales
• The Delta-nabla Calculus of Variations for Composition Functionals on Time Scales
• A symmetric Norlund sum with application to inequalities

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• Variational Calculus with Conformable Fractional Derivatives
• Some inequalities for interval-valued functions on time scales
• Existence and Uniqueness of Solution for a Fractional Riemann-Liouville Initial Value Problem on Time Scales
• Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales
• Chain rules and inequalities for the BHT fractional calculus on arbitrary time scales
• Existence and uniqueness results for a fractional Riemann-Liouville nonlocal thermistor problem on arbitrary time scales
• Generalized Fractional Operators on Time Scales with Application to Dynamic Equations
• Symmetric duality for left and right Riemann-Liouville and Caputo fractional differences
• Time-Fractional Optimal Control of Initial Value Problems on Time Scales
• Complex-valued fractional derivatives on time scales
• Variable order Mittag-Leffler fractional operators on isolated time scales and application to the calculus of variations
• Analysis of fractional integro-differential equations of thermistor type
• The fuzzy Henstock-Kurzweil delta integral on time scales