1,032 research outputs found
A simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications
AbstractIn many recent works, several authors demonstrated the usefulness of fractional calculus operators in the derivation of (explicit) particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how this simple fractional-calculus approach to the solutions of the classical Bessel differential equation of general order would lead naturally to several interesting consequences which include (for example) an alternative investigation of the power-series solutions obtainable usually by the Frobenius method. The methodology presented here is based largely upon some of the general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations
Fractional Bosonic Strings
The aim of this paper is to present a simple generalization of bosonic string
theory in the framework of the theory of fractional variational problems.
Specifically, we present a fractional extension of the Polyakov action, for
which we compute the general form of the equations of motion and discuss the
connection between the new fractional action and a generalization the
Nambu-Goto action. Consequently, we analyse the symmetries of the modified
Polyakov action and try to fix the gauge, following the classical procedures.
Then we solve the equations of motion in a simplified setting. Finally, we
present an Hamiltonian description of the classical fractional bosonic string
and introduce the fractional light-cone gauge. It is important to remark that,
throughout the whole paper, we thoroughly discuss how to recover the known
results as an "integer" limit of the presented model.Comment: 21 pages, no figure
Fractional Klein-Gordon equations and related stochastic processes
This paper presents finite-velocity random motions driven by fractional
Klein-Gordon equations of order . A key tool in the analysis
is played by the McBride's theory which converts fractional hyper-Bessel
operators into Erdelyi-Kober integral operators. Special attention is payed to
the fractional telegraph process whose space-dependent distribution solves a
non-homogeneous fractional Klein-Gordon equation. The distribution of the
fractional telegraph process for coincides with that of the
classical telegraph process and its driving equation converts into the
homogeneous Klein-Gordon equation. Fractional planar random motions at finite
velocity are also investigated, the corresponding distributions obtained as
well as the explicit form of the governing equations. Fractionality is
reflected into the underlying random motion because in each time interval a
binomial number of deviations (with uniformly-distributed
orientation) are considered. The parameter of is itself a
random variable with fractional Poisson distribution, so that fractionality
acts as a subsampling of the changes of directions. Finally the behaviour of
each coordinate of the planar motion is examined and the corresponding
densities obtained. Extensions to -dimensional fractional random flights are
envisaged as well as the fractional counterpart of the Euler-Poisson-Darboux
equation to which our theory applies
Bifurcation and Chaos in Fractional-Order Systems
This book presents a collection of seven technical papers on fractional-order complex systems, especially chaotic systems with hidden attractors and symmetries, in the research front of the field, which will be beneficial for scientific researchers, graduate students, and technical professionals to study and apply. It is also suitable for teaching lectures and for seminars to use as a reference on related topics
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