Fractional Generalization of Gradient Systems

  title={Fractional Generalization of Gradient Systems},
  author={Vasily E. Tarasov},
  journal={Letters in Mathematical Physics},
  • V. E. Tarasov
  • Published 1 July 2005
  • Mathematics
  • Letters in Mathematical Physics
We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these systems. 
Fractional-order Variational Derivative
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Fractional gradient and its application to the fractional advection equation
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Nonholonomic constraints with fractional derivatives
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Fractional analysis is applied to describe classical dynamical systems. Fractional derivative can be defined as a fractional power of derivative. The infinitesimal generators {H, �} and L = G(q, p)@q
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Differential forms and exterior calculus are important theories in mathematics. Exterior calculus have found wide applications in fields such as general relativity, theory of electromagnetic fields,
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Fractional generalization of Liouville equations.
Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition, which is considered as anormalization condition for systems in fractional phase space.
Fractional systems and fractional Bogoliubov hierarchy equations.
  • V. E. Tarasov
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2005
This work considers the fractional generalizations of the phase volume, volume element, and Poisson brackets, and fractional analogs of the Hamilton equations, which lead to an analog of thephase space and systems on this fractional phase space.
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Fractional Integrals and Derivatives: Theory and Applications
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Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with
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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.
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It is proved that the fractional integrals can be used to describe the media with noninteger mass dimensions and the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives are derived.