Fractional Differential Forms

Abstract

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case. Coordinate transformation rules are also computed. The transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative. The metric for the fractional form spaces is given, based on the coordinate transformation rules. All results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one.

20 pages

References in corpus (1)

• Fractional Differential Forms

Cited by in corpus (37)

• Fractional Vector Calculus and Fractional Maxwell's Equations
• Quantum field theory, gravity and cosmology in a fractal universe
• Geometry and field theory in multi-fractional spacetime
• Fractional Generalization of Gradient and Hamiltonian Systems
• Fractional Generalization of Liouville Equations
• Fractional Differential Forms
• Fractional Systems and Fractional Bogoliubov Hierarchy Equations
• Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches
• Fractional Liouville and BBGKI Equations
• Multi-scale gravity and cosmology
• Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory
• Adaptive molecular resolution via a continuous change of the phase space dimensionality
• Fractional Statistical Mechanics
• Nonholonomic Constraints with Fractional Derivatives
• Fractional Generalization of Gradient Systems
• Liouville and Bogoliubov Equations with Fractional Derivatives
• Symmetries and propagator in multifractional scalar field theory
• Generalized diffusion equation with fractional derivatives within Renyi statistics
• Theory on linear L-fractional differential equations and a new Mittag-Leffler-type function
• The geometry of fractional osculator bundle of higher order and applications
• Generalized Tu Formula and Hamilton Structures of Fractional Soliton Equation Hierarchy
• On the fractional damped oscillators and fractional forced oscillators
• Electro-chemical manifestation of nanoplasmonics in fractal media
• Fractional-Order Shell Theory: Formulation and Application to the Analysis of Nonlocal Cylindrical Panels
• Fractional-order Variational Derivative
• Generalized diffusion equation with nonlocality of space-time. Memory function modelling
• Advanced manifold-metric pairs
• Fractional Dynamical Systems on Fractional Leibniz Algebroids
• Geometry of fractional spaces
• Generalized diffusion equation with nonlocality of space-time: analytical and numerical analysis
• Fractional dynamical systems and applications in mechanics and economics
• Spatial Rotation of the Fractional Derivative in Two Dimensional Space
• Extending discrete exterior calculus to a fractional derivative
• A Comparison of Constitutive Models of Blood
• On a Leibnitz-type fractional derivative
• Matrix Order Differintegration
• Derivatives and integrals: matrix order operators as an extension of the fractional calculus