425 research outputs found
Fractional variational calculus for nondifferentiable functions
We prove necessary optimality conditions, in the class of continuous
functions, for variational problems defined with Jumarie's modified
Riemann-Liouville derivative. The fractional basic problem of the calculus of
variations with free boundary conditions is considered, as well as problems
with isoperimetric and holonomic constraints.Comment: Submitted 13-Aug-2010; revised 24-Nov-2010; accepted 28-March-2011;
for publication in Computers and Mathematics with Application
Oscillation of Non-Linear Systems Close to Equilibrium Position in the Presence of Coarse-Graining in Time and Space
One considers the motion of nonlinear systems close to their equilibrium positions in the presence of coarse-graining in time on the one hand, and coarse-graining in time on the other hand. By considering a coarse-grained time as a time in which the increment is not dt but rather (dt)c > dt, one is led to introduce a modeling in terms of fractional derivative with respect to time; and likewise for coarse-graining with respect to the space variable x. After a few prerequisites on fractional calculus via modified Riemann-Liouville derivative, one examines in a detailed way the solutions of fractional linear differential equations in this framework, and then one uses this result in the linearization of nonlinear systems close to their equilibrium positions
Extending the D'Alembert Solution to Space-Time Modified Riemann-Liouville Fractional Wave Equations
In the realm of complexity, it is argued that adequate modeling of
TeV-physics demands an approach based on fractal operators and fractional
calculus (FC). Non-local theories and memory effects are connected to
complexity and the FC. The non-differentiable nature of the microscopic
dynamics may be connected with time scales. Based on the Modified
Riemann-Liouville definition of fractional derivatives, we have worked out
explicit solutions to a fractional wave equation with suitable initial
conditions to carefully understand the time evolution of classical fields with
a fractional dynamics. First, by considering space-time partial fractional
derivatives of the same order in time and space, a generalized fractional
D'Alembertian is introduced and by means of a transformation of variables to
light-cone coordinates, an explicit analytical solution is obtained. To address
the situation of different orders in the time and space derivatives, we adopt
different approaches, as it will become clear throughout the paper. Aspects
connected to Lorentz symmetry are analyzed in both approaches.Comment: 8 page
Generalized Tu Formula and Hamilton Structures of Fractional Soliton Equation Hierarchy
With the modified Riemann-Liouville fractional derivative, a fractional Tu
formula is presented to investigate generalized Hamilton structure of
fractional soliton equations. The obtained results can be reduced to the
classical Hamilton hierachy of ordinary calculus.Comment: 12 p
Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations
Recently, fractional differential equations have been investigated via the
famous variational iteration method. However, all the previous works avoid the
term of fractional derivative and handle them as a restricted variation. In
order to overcome such shortcomings, a fractional variational iteration method
is proposed. The Lagrange multipliers can be identified explicitly based on
fractional variational theory.Comment: 12 pages, 1 figure
Emergence of order in selection-mutation dynamics
We characterize the time evolution of a d-dimensional probability
distribution by the value of its final entropy. If it is near the
maximally-possible value we call the evolution mixing, if it is near zero we
say it is purifying. The evolution is determined by the simplest non-linear
equation and contains a d times d matrix as input. Since we are not interested
in a particular evolution but in the general features of evolutions of this
type, we take the matrix elements as uniformly-distributed random numbers
between zero and some specified upper bound. Computer simulations show how the
final entropies are distributed over this field of random numbers. The result
is that the distribution crowds at the maximum entropy, if the upper bound is
unity. If we restrict the dynamical matrices to certain regions in matrix
space, for instance to diagonal or triangular matrices, then the entropy
distribution is maximal near zero, and the dynamics typically becomes
purifying.Comment: 8 pages, 8 figure
Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
We generalize the fractional variational problem by allowing the possibility
that the lower bound in the fractional derivative does not coincide with the
lower bound of the integral that is minimized. Also, for the standard case when
these two bounds coincide, we derive a new form of Euler-Lagrange equations. We
use approximations for fractional derivatives in the Lagrangian and obtain the
Euler-Lagrange equations which approximate the initial Euler-Lagrange equations
in a weak sense
Entropic trade-off relations for quantum operations
Spectral properties of an arbitrary matrix can be characterized by the
entropy of its rescaled singular values. Any quantum operation can be described
by the associated dynamical matrix or by the corresponding superoperator. The
entropy of the dynamical matrix describes the degree of decoherence introduced
by the map, while the entropy of the superoperator characterizes the a priori
knowledge of the receiver of the outcome of a quantum channel Phi. We prove
that for any map acting on a N--dimensional quantum system the sum of both
entropies is not smaller than ln N. For any bistochastic map this lower bound
reads 2 ln N. We investigate also the corresponding R\'enyi entropies,
providing an upper bound for their sum and analyze entanglement of the
bi-partite quantum state associated with the channel.Comment: 10 pages, 4 figure
Random-time processes governed by differential equations of fractional distributed order
We analyze here different types of fractional differential equations, under
the assumption that their fractional order is random\ with
probability density We start by considering the fractional extension
of the recursive equation governing the homogeneous Poisson process
\ We prove that, for a particular (discrete) choice of , it
leads to a process with random time, defined as The distribution of the
random time argument can be
expressed, for any fixed , in terms of convolutions of stable-laws. The new
process is itself a renewal and
can be shown to be a Cox process. Moreover we prove that the survival
probability of , as well as its
probability generating function, are solution to the so-called fractional
relaxation equation of distributed order (see \cite{Vib}%).
In view of the previous results it is natural to consider diffusion-type
fractional equations of distributed order. We present here an approach to their
solutions in terms of composition of the Brownian motion with the
random time . We thus provide an
alternative to the constructions presented in Mainardi and Pagnini
\cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order
case.Comment: 26 page
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