A Fractional Lie Group Method For Anomalous Diffusion Equations

Anomalous Rotational Relaxation: A Fractional Fokker-Planck Equation Approach

In this study we obtained analytically relaxation function in terms of rotational correlation functions based on Brownian motion for complex disordered systems in a stochastic framework. We found out that rotational relaxation function has a fractional form for complex disordered systems, which indicates relaxation has non-exponential character obeys to Kohlrausch-William-Watts law, following the Mittag-Leffler decay.

Solutions For A Generalized Fractional Anomalous Diffusion Equation

In this paper, we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time-dependent diffusion coefficient and an external force into account, which subjects to the natural boundaries and the generic initial condition. We obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the Tsallis entropy.

Anomalous relaxation in dielectrics with Hilfer fractional derivative

We introduce a new relaxation function depending on an arbitrary parameter as solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order ξ, with 0<ξ<1 and type η, with 0<η<1. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a b…

Fractional derivatives: Fourier, elephants, memory effects, viscoelastic materials and anomalous diffusions

This paper, that will appear in the Notices of the AMS, begins with a brief historical account of the beginnings of fractional calculus and the crucial roles played by Leibniz and Fourier. Fourier's definition of fractional derivative is introduced and unpacked by using the modern technique known as the method of semigroups. Recent advances on the theory of fractional derivatives are presented. Furthermore, we address some questions that have been raised by some in the scientific community. Finally, we present three different applications: population growth with memory, viscoleastic materials …

Lie Symmetry Analysis of Some Conformable Fractional Partial Differential Equations

In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg-de Vries, modified Korteweg-de Vries, Burgers, and modified Burgers equations with conformable fractional time and space derivatives. For each equation, all of the vector fields and the Lie symmetries are obtained. Moreover, exact solutions are given to these equations in terms of solutions of ordinary differential equations. In particular, it is shown that the …

A note on paper "Anomalous relaxation model based on the fractional derivative with a Prabhakarlike kernel" [Z. Angew. Math. Phys. (2019) 70:42]

Inspired by the article "Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel" (Z. Angew. Math. Phys. (2019) 70:42) which authors D. Zhao and HG. Sun studied the integro-differential equation with the kernel given by the Prabhakar function $e^{-γ}_{α, β}(t, λ)$ we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times which admissible range extends the limits given in \cite[Theorem 3.1]{DZhao2019} to all positive values. For special choices of parameters entering the equ…

The time-fractional radiative transport equation -- Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion

We consider the radiative transport equation in which the time derivative is replaced by the Caputo derivative. Such fractional-order derivatives are related to anomalous transport and anomalous diffusion. In this paper we describe how the time-fractional radiative transport equation is obtained from continuous-time random walk and see how the equation is related to the time-fractional diffusion equation in the asymptotic limit. Then we solve the equation with Legendre-polynomial expansion.

Time-space fabric underlying anomalous diffusion

This study unveils the time-space transforms underlying anomalous diffusion process. Based on this finding, we present the two hypotheses concerning the effect of fractal time-space fabric on physical behaviors and accordingly derive fractional quantum relationships between energy and frequency, momentum and wavenumber which further give rise to fractional Schrodinger equation. As an alternative modeling approach to the standard fractional derivatives, we introduce the concept of the Hausdorff derivative underlying the Hausdorff dimensions of metric spacetime. And in terms of the proposed hypo…

Anomalous Topological Pumps and Fractional Josephson Effects

We discover novel topological pumps in the Josephson effects for superconductors. The phase difference, which is odd under the chiral symmetry defined by the product of time-reversal and particle-hole symmetries, acts as an anomalous adiabatic parameter. These pumping cycles are different from those in the "periodic table", and are characterized by $Z\times Z$ or $Z_2\times Z_2$ strong invariants. We determine the general classifications in class AIII, and those in class DIII with a single anomalous parameter. For the $Z_2\times Z_2$ topological pump in class DIII, one $Z_2$ invariant describe…

识别分数量子反常霍尔系统中的任意子拓扑序

Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems

该研究通过代数拓扑方法确定分数量子反常霍尔系统中任意子的结构，揭示了其与陈数相关的编织相位特性，为理解拓扑序提供了理论基础。

Fractional diffusion equation description of an open anomalous heat conduction set-up

We provide a stochastic fractional diffusion equation description of energy transport through a finite one-dimensional chain of harmonic oscillators with stochastic momentum exchange and connected to Langevian type heat baths at the boundaries. By establishing an unambiguous finite domain representation of the associated fractional operator, we show that this equation can correctly reproduce equilibrium properties like Green-Kubo formula as well as non-equilibrium properties like the steady state temperature and current. In addition, this equation provides the exact time evolution of the tempe…

Anomalous Diffusion in One-Dimensional Disordered Systems: A Discrete Fractional Laplacian Method

This work extends the applications of Anderson-type Hamiltonians to include transport characterized by anomalous diffusion. Herein, we investigate the transport properties of a one-dimensional disordered system that employs the discrete fractional Laplacian, $(-Δ)^s,\ s\in(0,2),$ in combination with results from spectral and measure theory. It is a classical mathematical result that the standard Anderson model exhibits localization of energy states for all nonzero disorder in one-dimensional systems. Numerical simulations utilizing our proposed model demonstrate that this localization effect i…

Fractional Super Lie Algebras and Groups

n^{th} root of a Lie algebra and its dual (that is fractional supergroup) based on the permutation group $S_n$ invariant forms are formulated in the Hopf algebra formalism. Detailed discussion of $S_3$-graided $sl(2)$ algebras is done.

反常随机飞行与时间分数阶跑动-翻滚方程

Anomalous random flights and time-fractional run-and-tumble equations

本文将d维随机飞行的一般性分析扩展至由动力学方程非局部广义化产生的时间分数阶过程，通过时间分数阶方程解的概率解释导出原始输运过程的时间变换版本，并重点研究了一维电报过程的时间分数阶方程及其随机解。

The role of fractional time-derivative operators on anomalous diffusion

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional…

Quantum mechanics in fractional and other anomalous spacetimes

We formulate quantum mechanics in spacetimes with real-order fractional geometry and more general factorizable measures. In spacetimes where coordinates and momenta span the whole real line, Heisenberg's principle is proven and the wave-functions minimizing the uncertainty are found. In spite of the fact that ordinary time and spatial translations are broken and the dynamics is not unitary, the theory is in one-to-one correspondence with a unitary one, thus allowing us to employ standard tools of analysis. These features are illustrated in the examples of the free particle and the harmonic osc…

A foundational approach to the Lie theory for fractional order partial differential equations

We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

Fractional Supersymmetry and Lie Algebras

Supersymmetry and super-Lie algebras have been consistently generalized previously. The so-called fractional supersymmetry and $F-$Lie algebras could be constructed starting from any representation $\D$ of any Lie algebra $g$. This involves taking the $F^{\mathrm th}$ root of $\D$ in some sense. We show, after having constructed differential realization(s) of any Lie algebra, how fractional supersymmetry can be explicitly realized in terms of appropriate homogeneous monomials.

(2+1)维时间分数阶Kadomtsev-Petviashvili方程的Lie对称分析

Lie symmetry analysis of (2+1)-dimensional time fractional Kadomtsev-Petviashvili equation

本文利用Lie对称分析研究(2+1)维时间分数阶KP方程，通过对称约化得到低维分数阶方程及其精确解，并构造了方程的守恒律。

Fractional Supersymmetry and Infinite Dimensional Lie Algebras

In an earlier work extensions of supersymmetry and super Lie algebras were constructed consistently starting from any representation $\D$ of any Lie algebra $\g$. Here it is shown how infinite dimensional Lie algebras appear naturally within the framework of fractional supersymmetry. Using a differential realization of $\g$ this infinite dimensional Lie algebra, containing the Lie algebra $\g$ as a sub-algebra, is explicitly constructed.

Fractional Supersymmetry and F-fold Lie Superalgebras

We give infinite dimensional and finite dimensional examples of $F-$fold Lie superalgebras. The finite dimensional examples are obtained by an inductive procedure from Lie algebras and Lie superalgebras.

Cohomological resolutions for anomalous Lie constraints

It is shown that the BRST resolution of the spaces of physical states of the systems with anomalies can be consistently defined. The appropriate anomalous complexes are obtained by canonical restrictions of the ghost extended spaces to the kernel of anomaly operator without any modifications of the "matter" sector. The cohomologies of the anomalous complex for the case of constarints constituting a centrally extended simple Lie algebra of compact type are calculated and analyzed in details within the framework of Hodge - deRham - Kaehler theory: the vanishing theorem of the relative cohomologi…

Generalized distributed order diffusion equations with composite time fractional derivative

In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier-Laplace transform method. We represent solutions in terms of infinite series in Fox $H$-functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obt…

Accelerating and Retarding Anomalous Diffusion

In this paper Gaussian models of retarded and accelerated anomalous diffusion are considered. Stochastic differential equations of fractional order driven by single or multiple fractional Gaussian noise terms are introduced to describe retarding and accelerating subdiffusion and superdiffusion. Short and long time asymptotic limits of the mean squared displacement of the stochastic processes associated with the solutions of these equations are studied. Specific cases of these equations are shown to provide possible descriptions of retarding or accelerating anomalous diffusion.

Numerical simulations of anomalous diffusion

In this paper we present numerical methods - finite differences and finite elements - for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is a phenomenon connected with the interactions within the complex and non-homogeneous background. In order to consider physical initial-value conditions we use fractional derivative in the Caputo sense. In numerical analysis the boundary conditions of first kind are accounted and in the final part of this paper the result of simulations are presented.

含变系数的时间分数电报系统的对称分类与不变解

Lie symmetry classification and invariant solutions of time-fractional telegraph systems with variable coefficients

本文研究了含变系数的时间分数电报系统的对称分类，通过李对称分析和Riemann-Liouville分数导数，确定了系数函数的对称扩展形式，并获得闭合形式的不变解。

tempered分数阶Keller-Segel系统的李对称分析与相似约化

Lie symmetry analysis and similarity reductions for the tempered-fractional Keller Segel system

本文对含反常扩散的趋化模型tempered分数阶Keller-Segel系统开展李对称分析，用新方法处理其分数阶算子非局域性，将其PDE约化为ODE得到新精确解，所得结果为该模型行为研究提供见解并提出通用方法。

Lie Algebra and Conservation Laws for the Time-fractional Heat Equation

The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that that the numbers of symmetries and Lie brackets are reduced significantly as compared to the nonfractional order for all the dimensions. In fact for integer order linear heat equation the number of solution symmetries is equal to the product of the order and space dimension, whereas for the fractional case, it is half of the product on the order and space dimension. We have classified the symmetries and discussed the Lie algebras and conservational laws. We generalise …

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=…

Lie symmetry analysis of a class of time fractional nonlinear evolution systems

We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. We find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain. In each case, the dimension of the Lie algebra generated by the infinitesimal symmetries is greater than 2, and for this reason we present the structu…

Transient anomalous sub-diffusion on bounded domains

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and eigenfunction expansions in time and space, are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator.

On G-fractional diffusion models in bounded domains

In the recent literature, the g-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive regimes. In this paper we study the problem of g-fractional diffusion in a bounded domain with absorbing boundaries. We find the explicit solution for the initial-boundary value problem and we study the first passage time distribution and the mean first-passage time (MFPT). An interestin outcome is the proof that with a particular choice of the function $g$ …

Brownian motion and anomalous diffusion revisited via a fractional Langevin equation

In this paper we revisit the Brownian motion on the basis of {the fractional Langevin equation which turns out to be a particular case of the generalized Langevin equation introduced by Kubo in 1966. The importance of our approach is to model the Brownian motion more realistically than the usual one based on the classical Langevin equation, in that it takes into account also the retarding effects due to hydrodynamic back-flow, i.e. the added mass and the Basset memory drag. We provide the analytical expressions of the correlation functions (both for the random force and the particle velocity) …

Characterizations and simulations of a class of stochastic processes to model anomalous diffusion

In this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusion. This class, called generalized grey Brownian motion (ggBm), is made up off self-similar with stationary increments processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and beta=1, and time-fractional diffusion stochastic processes when alpha=beta in (0,1). The latters have marginal probability density function governed by time-fractional diffusion equations of order beta. The ggBm is defi…