Journal ArticleDOI
Fractional master equation: non-standard analysis and Liouville–Riemann derivative
TLDR
In this article, the difference between Liouville-Riemann fractional derivatives and non-standard analysis of fractional Poisson processes is discussed. But the present paper only considers Poisson process models with long-range dependence.Abstract:
Fractional master equations may be defined either by means of Liouville–Riemann (L–R) fractional derivative or via non-standard analysis. The first approach describes processes with long-range dependence whilst the second approach deals with processes involving independent increments. The present papers put in evidence some of the differences between these two modellings, and to this end it especially considers more fractional Poisson processes.read more
Citations
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Journal ArticleDOI
The Fractional Poisson Process and the Inverse Stable Subordinator
TL;DR: In this article, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times, which unifies the two main approaches in stochastic theory of time-fractional diffusion equations.
Journal ArticleDOI
Fractional Poisson processes and related planar random motions
Luisa Beghin,Enzo Orsingher +1 more
TL;DR: In this article, three different fractional versions of the standard Poisson process and some related results concerning the distribution of order statistics and the compound poisson process are presented, and a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by fractional Poisson processes is presented.
Journal ArticleDOI
Control of a class of fractional-order chaotic systems via sliding mode
TL;DR: This paper investigates the chaos control of a class of fractional-order chaotic systems via sliding mode through the derived sliding mode control law, and guarantees asymptotical stability of the uncertain fractionsal- order chaotic systems in the presence of an external disturbance.
Journal ArticleDOI
Chaos in a fractional order modified Duffing system
Zheng-Ming Ge,Chan-Yi Ou +1 more
TL;DR: In this paper, the chaotic behaviors in a fractional order modified Duffing system are studied numerically by phase portraits, Poincare maps and bifurcation diagrams, and linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in (0, 1), based on frequency domain arguments.
Journal ArticleDOI
Variational problems with fractional derivatives: Invariance conditions and Nöther’s theorem☆
TL;DR: In this paper, a variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases: necessary and sufficient conditions for an infinitesimal transformation group (basic Nother's identity) are obtained.
References
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Book
Fractal Space-Time And Microphysics: Towards A Theory Of Scale Relativity
TL;DR: In this paper, a general introduction from fractal objects to fractal spaces fractal dimension of a quantum path the fractal structure of the quantum space-time towards a linear theory of scale relativity prospects.
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From continuous time random walks to the fractional fokker-planck equation
TL;DR: The domain of validity of the fractional kinetic equation is discussed, and the CTRW solution and that of the FFPE are compared for the force free case.
Journal ArticleDOI
Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach
TL;DR: In this paper, a fractional Fokker-Planck equation describing the stochastic evolution of a particle under the combined influence of an external, nonlinear force and a thermal heat bath is introduced.
Journal ArticleDOI
Fractional master equations and fractal time random walks.
TL;DR: It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks for fractional time derivatives of order 0.1.
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Statistical dynamics: matter out of equilibrium
TL;DR: In this paper, a general formalism of statistical mechanics reduced distribution functions and correlation functions is proposed. But the model is restricted to the mean field approximation weak coupling kinetic equation for dilute gases and the properties of kinetic equations hydrodynamics and tranport transport.