Fractional Electromagnetic Equations Using Fractional Forms

The generalized physics laws involving fractional derivatives give new models and conceptions that can be used in complex systems having memory effects. Using the fractional differential forms, the classical electromagnetic equations involving the fractional derivatives have been worked out. The fractional conservation law for the electric charge and the wave equations were derived by using this method. In addition, the fractional vector and scalar potentials and the fractional Poynting theorem have been derived.     

Riesz Fractional Derivatives and Fractional Dimensional Space

The Fourier transform method is used to solve fractional Poisson’s equation with Riesz fractional derivative of order α. It is shown that the solution is given in terms of the fractional dimensional space D. Gauss law for the electrostatic problem is given and the total electric flux is obtained in terms of α and D.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.     

Fractional motions

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.     

Fractional Number Operator and Associated Fractional Diffusion Equations

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.     

Fractional Noether Theorem Based on Extended Exponentially Fractional Integral

Based on the new type of fractional integral definition, namely extended exponentially fractional integral introduced by EI-Nabulsi, we study the fractional Noether symmetries and conserved quantities for both holonomic system and nonholonomic system. First, the fractional variational problem under the sense of extended exponentially fractional integral is established, the fractional d’Alembert-Lagrange principle is deduced, then the fractional Euler-Lagrange equations of holonomic system and the fractional Routh equations of nonholonomic system are given; secondly, the invariance of fractional Hamilton action under infinitesimal transformations of group is also discussed, the corresponding definitions and criteria of fractional Noether symmetric transformations and quasi-symmetric transformations are established; finally, the fractional Noether theorems for both holonomic system and nonholonomic system are explored. What’s more, the relationship between the fractional Noether symmetry and conserved quantity are revealed.     

联合分数变换相关器的分数相关特性分析

从分数傅里叶变换（FRFT）的定义出发，理论分析了联合分数变换相关器（JFRTC）的分数相关特性.从所得JFRTC的数学表达式中可以看出，将FRFT应用到联合变换相关器（JTC）中得到的JFRTC具有与传统JTC不同的性质.对于传统JTC，一旦输入平面上参考图像与目标图像之间的距离给定，相关输出峰的位置即确定，而JFRTC的相关输出峰的位置则可以由分数级次p₁和p₂来自由调节，这个特性在实际模式识别中非常有用.另一方面，JFRTC的相关输出峰值在大多数情况下低于传统JTC的相关峰值，却是JFRTC的一大缺点.最后，从FRFT的比例性质出发，给出了FRFT谱畸变不变的实现条件，并由此预言了JFRTC畸变不变模式识别的功能.     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

Fractional vortex lens

The properties of a fractional vortex lens are discussed and its focal plane intensity distribution is investigated. The azimuthal phase pattern of a fractional topological charge vortex and the quadratic phase variation of the lens are displayed on a spatial light modulator (SLM) that is used in the phase mode. The intensity distribution possesses a low intensity radial opening in contrast to the doughnut pattern formed by a vortex beam with integer charge. The presence of multiple singularities in the radial opening of the doughnut structure is detected interferometrically.     

Fractional optical solitons

This Letter shows that soliton propagation can be described by an extended NLS equation which incorporates fractional dispersion and a fractional nonlinearity. The fractional dispersive term is written in terms of Grünwald-Letnikov fractional derivatives (FDs). Forward and backward FDs are introduced in order to satisfy the conservation of energy. It is found that the soliton solutions of this model form a continuous family and are stable. The Vakhitov-Kolokolov criterion is used to confirm the stability of these fractional solitons.     

Fractional standard map

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the map parameter and the fractional order of the derivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points, stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.     

Fractional statistical mechanics

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Liouville and Bogoliubov equations with fractional coordinate and momenta derivatives are considered as a basis to derive fractional kinetic equations. The Fokker-Planck-Zaslavsky equation that has fractional phase-space derivatives is obtained from the fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.     

Uncovering Fractional Monodromy

The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n ₁:(?n ₂) resonant Hamiltonian systems with n ₁, n ₂ coprime natural numbers. We consider, in particular, systems that for n ₁, n ₂ > 1 contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n ₁, n ₂. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.     

Fractional Nambu Mechanics

The fractional generalization of Nambu mechanics is constructed by using the differential forms and exterior derivatives of fractional orders. The generalized Pfaffian equations are obtained and one example is investigated in details. On leave of absence from Institute of Space Sciences, P.O. Box, MG-23, 76900, Magurele-Bucharest, Romania.     

Fractional Gabor transform

A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.     

Fractional quiver W-algebras

We introduce quiver gauge theory associated with the non-simply laced type fractional quiver and define fractional quiver W-algebras by using construction of Kimura and Pestun (Lett Math Phys, 2018.  https://doi.org/10.1007/s11005-018-1072-1; Lett Math Phys, 2018.  https://doi.org/10.1007/s11005-018-1073-0) with representation of fractional quivers.