1,280 research outputs found
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
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
A Fractional Lie Group Method For Anomalous Diffusion Equations
Lie group method provides an efficient tool to solve a differential equation.
This paper suggests a fractional partner for fractional partial differential
equations using a fractional characteristic method. A space-time fractional
diffusion equation is used as an example to illustrate the effectiveness of the
Lie group method.Comment: 5 pages,in pres
Anomalous g-Factors for Charged Leptons in a Fractional Coarse-Grained Approach
In this work, we investigate aspects of the electron, muon and tau
gyromagnetic ratios (g-factor) in a fractional coarse-grained scenario, by
adopting a Modified Riemann-Liouville (MRL) fractional calculus. We point out
the possibility of mapping the experimental values of the specie's g-factors
into a theoretical parameter which accounts for fractionality, without
computing higher-order QED calculations. We wish to understand whether the
value of (g-2) may be traced back to a fractionality of space-time.The
justification for the difference between the experimental and the theoretical
value g=2 stemming from the Dirac equation is given in the terms of the
complexity of the interactions of the charged leptons, considered as
pseudo-particles and "dressed" by the interactions and the medium. Stepwise, we
build up a fractional Dirac equation from the fractional Weyl equation that, on
the other hand, was formulated exclusively in terms of the helicity operator.
From the fractional angular momentum algebra, in a coarse-grained scenario, we
work out the eigenvalues of the spin operator. Based on the standard
electromagnetic current, as an analogy case, we write down a fractional
Lagrangian density, with the electromagnetic field minimally coupled to the
particular charged lepton. We then study a fractional gauge-like invariance
symmetry, formulate the covariant fractional derivative and propose the spinor
field transformation. Finally, by taking the non-relativistic regime of the
fractional Dirac equation, the fractional Pauli equation is obtained and, from
that, an explicit expression for the fractional g-factor comes out that is
compared with the experimental CODATA value. Our claim is that the different
lepton species must probe space-time by experiencing different fractionalities,
once the latter may be associated to the effective interactions of the
different families with the medium.Comment: 15 page
Time-Fractional KdV Equation: Formulation and Solution using Variational Methods
In this work, the semi-inverse method has been used to derive the Lagrangian
of the Korteweg-de Vries (KdV) equation. Then, the time operator of the
Lagrangian of the KdV equation has been transformed into fractional domain in
terms of the left-Riemann-Liouville fractional differential operator. The
variational of the functional of this Lagrangian leads neatly to Euler-Lagrange
equation. Via Agrawal's method, one can easily derive the time-fractional KdV
equation from this Euler-Lagrange equation. Remarkably, the time-fractional
term in the resulting KdV equation is obtained in Riesz fractional derivative
in a direct manner. As a second step, the derived time-fractional KdV equation
is solved using He's variational-iteration method. The calculations are carried
out using initial condition depends on the nonlinear and dispersion
coefficients of the KdV equation. We remark that more pronounced effects and
deeper insight into the formation and properties of the resulting solitary wave
by additionally considering the fractional order derivative beside the
nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure
Weyl and Marchaud derivatives: a forgotten history
In this paper we recall the contribution given by Hermann Weyl and Andr\'e
Marchaud to the notion of fractional derivative. In addition we discuss some
relationships between the fractional Laplace operator and Marchaud derivative
in the perspective to generalize these objects to different fields of the
mathematics.Comment: arXiv admin note: text overlap with arXiv:1705.00953 by other author
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