1,127 research outputs found
Fractional diffusions with time-varying coefficients
This paper is concerned with the fractionalized diffusion equations governing
the law of the fractional Brownian motion . We obtain solutions of
these equations which are probability laws extending that of . Our
analysis is based on McBride fractional operators generalizing the hyper-Bessel
operators and converting their fractional power into
Erd\'elyi--Kober fractional integrals. We study also probabilistic properties
of the r.v.'s whose distributions satisfy space-time fractional equations
involving Caputo and Riesz fractional derivatives. Some results emerging from
the analysis of fractional equations with time-varying coefficients have the
form of distributions of time-changed r.v.'s
Diffusion in quantum geometry
The change of the effective dimension of spacetime with the probed scale is a
universal phenomenon shared by independent models of quantum gravity. Using
tools of probability theory and multifractal geometry, we show how dimensional
flow is controlled by a multiscale fractional diffusion equation, and
physically interpreted as a composite stochastic process. The simplest example
is a fractional telegraph process, describing quantum spacetimes with a
spectral dimension equal to 2 in the ultraviolet and monotonically rising to 4
towards the infrared. The general profile of the spectral dimension of the
recently introduced multifractional spaces is constructed for the first time.Comment: 5 pages, 1 figure. v2: title slightly changed, discussion improve
Fractional Curve Flows and Solitonic Hierarchies in Gravity and Geometric Mechanics
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler
spaces are applied in fractional calculus with Caputo derivatives and for
elaborating models of fractional gravity and fractional Lagrange mechanics. The
geometric data for such models are encoded into (fractional) bi-Hamiltonian
structures and associated solitonic hierarchies. The constructions yield
horizontal/vertical pairs of fractional vector sine-Gordon equations and
fractional vector mKdV equations when the hierarchies for corresponding curve
fractional flows are described in explicit forms by fractional wave maps and
analogs of Schrodinger maps.Comment: latex2e, 11pt, 21 pages; the variant accepted to J. Math. Phys.; new
and up--dated reference
Discrete Map with Memory from Fractional Differential Equation of Arbitrary Positive Order
Derivatives of fractional order with respect to time describe long-term
memory effects. Using nonlinear differential equation with Caputo fractional
derivative of arbitrary order , we obtain discrete maps with
power-law memory. These maps are generalizations of well-known universal map.
The memory in these maps means that their present state is determined by all
past states with power-law forms of weights. Discrete map equations are
obtained by using the equivalence of the Cauchy-type problem for fractional
differential equation and the nonlinear Volterra integral equation of the
second kind
On the Consistency of the Solutions of the Space Fractional Schr\"odinger Equation
Recently it was pointed out that the solutions found in literature for the
space fractional Schr\"odinger equation in a piecewise manner are wrong, except
the case with the delta potential. We reanalyze this problem and show that an
exact and a proper treatment of the relevant integral proves otherwise. We also
discuss effective potential approach and present a free particle solution for
the space and time fractional Schr\"odinger equation in general coordinates in
terms of Fox's H-functions
Coupled systems of fractional equations related to sound propagation: analysis and discussion
In this note we analyse the propagation of a small density perturbation in a
one-dimensional compressible fluid by means of fractional calculus modelling,
replacing thus the ordinary time derivative with the Caputo fractional
derivative in the constitutive equations. By doing so, we embrace a vast
phenomenology, including subdiffusive, superdiffusive and also memoryless
processes like classical diffusions. From a mathematical point of view, we
study systems of coupled fractional equations, leading to fractional diffusion
equations or to equations with sequential fractional derivatives. In this
framework we also propose a method to solve partial differential equations with
sequential fractional derivatives by analysing the corresponding coupled system
of equations
Solution of Euler-Type Non-Homogeneous Differential Equations with Three Fractional Derivatives
The linear non-homogeneous ordinary differential equations with three left-hand sided Lioville derivatives of fractional order are considered. Using the direct and inverse Mellin transforms and the residue theory, explicit solutions of such equations are established in terms of the generalized hypergeometric Wright functions, of the generalized hypergeometric functions and of the Euler psi function. The corresponding results are deduced for ordinary differential equations of Euler type. Examples are given.
Mathematics Subject Classification: 34A05, 26A33, 44A99, 33C20, 33C99.
Key Words and Phrases: linear differential equations with Liouville fractional derivatives, ordinary differential equations, explicit solutions, Mellin transforms, generalized Wright function, generalized hypergeometric function, Euler psi function
Dynamics of the Chain of Oscillators with Long-Range Interaction: From Synchronization to Chaos
We consider a chain of nonlinear oscillators with long-range interaction of
the type 1/l^{1+alpha}, where l is a distance between oscillators and 0< alpha
<2. In the continues limit the system's dynamics is described by the
Ginzburg-Landau equation with complex coefficients. Such a system has a new
parameter alpha that is responsible for the complexity of the medium and that
strongly influences possible regimes of the dynamics. We study different
spatial-temporal patterns of the dynamics depending on alpha and show
transitions from synchronization of the motion to broad-spectrum oscillations
and to chaos.Comment: 22 pages, 10 figure
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