13,736 research outputs found
Fractional euler limits and their applications
© 2017 Society for Industrial and Applied Mathematics. Generalizations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An application to the Schlögl reactions with Mittag- Leffler waiting times is described
Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives
The classical fields with fractional derivatives are investigated by using
the fractional Lagrangian formulation.The fractional Euler-Lagrange equations
were obtained and two examples were studied.Comment: 9 page
The Method of almost convergence with operator of the form fractional order and applications
The purpose of this paper is twofold. First, basic concepts such as Gamma
function, almost convergence, fractional order difference operator and sequence
spaces are given as a survey character. Thus, the current knowledge about those
concepts are presented. Second, we construct the almost convergent spaces with
fractional order difference operator and compute dual spaces which are help us
in the characterization of matrix mappings. After we characterize to the matrix
transformations, we give some examples.Comment: 20 pages, 4 table
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
Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit
The 1D discrete fractional Laplacian operator on a cyclically closed
(periodic) linear chain with finitenumber of identical particles is
introduced. We suggest a "fractional elastic harmonic potential", and obtain
the -periodic fractionalLaplacian operator in the form of a power law matrix
function for the finite chain ( arbitrary not necessarily large) in explicit
form.In the limiting case this fractional Laplacian
matrix recovers the fractional Laplacian matrix ofthe infinite chain.The
lattice model contains two free material constants, the particle mass and
a frequency.The "periodic string continuum limit" of the
fractional lattice model is analyzed where lattice constant and
length of the chain ("string") is kept finite: Assuming finiteness of
the total mass and totalelastic energy of the chain in the continuum limit
leads to asymptotic scaling behavior for of thetwo material
constants,namely and . In
this way we obtain the -periodic fractional Laplacian (Riesz fractional
derivative) kernel in explicit form.This -periodic fractional Laplacian
kernel recovers for the well known 1D infinite space
fractional Laplacian (Riesz fractional derivative) kernel. When the scaling
exponentof the Laplacian takesintegers, the fractional Laplacian kernel
recovers, respectively, -periodic and infinite space (localized)
distributionalrepresentations of integer-order Laplacians.The results of this
paper appear to beuseful for the analysis of fractional finite domain problems
for instance in anomalous diffusion (L\'evy flights), fractional Quantum
Mechanics,and the development of fractional discrete calculus on finite
lattices
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