The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.     

Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases

Two model examples of the application of fractional calculus are considered. The Riemann–Liouville fractional derivative with 0 < α ≤ 1 was used. The solution of a fractional equation, which describes anomalous relaxation and diffusion in an isotropic fractal space, has been obtained in the form of the product of a Fox function by a Mittag-Leffler function. The solution is simpler than that given in Ref. 6 and it generalizes the result reported in Ref. 7. For the quantum case, a solution of the generalized Neumann–Kolmogorov fractional quantum-statistical equation has been obtained for an incomplete statistical operator which describes the random walk of a quantum spin particle, retarded in traps over a fractal space. The solution contains contributions from quantum Mittag-Leffler (nonharmonic) fractional oscillations, anomalous relaxation, noise fractional oscillations, and exponential fractional diffusion oscillation damping.     

Propagation characteristics of elegant Laguerre-Gaussian beams in the fractional Hankel transform plane

The fractional Hankel transform (FRHT) system is applied to study the transformation properties of elegant Laguerre-Gaussian beams (ELGBs). An analytical formula is derived for the FRHT of ELGBs based on the definition of the FRHB. By using the derived formula and the definition of the kurtosis parameter, the analytical propagation expression for the kurtosis parameter of ELGBs in the FRHT plane is derived. Some detailed numerical examples are given to illustrate the analytical results. The results show that the intensity distribution and the kurtosis parameter of elegant Laguerre-Gaussian beams in the fractional Hankel transform plane not only depend on order p and index m of Laguerre polynomial, but also evolve periodically with fractional order p of fractional system.     

Symmetries,Symmetry Reductions and Exact Solutions to the Generalized Nonlinear Fractional Wave Equations

In this paper, the Lie group classification method is performed on the fractional partial differential equation (FPDE), all of the point symmetries of the FPDEs are obtained. Then, the symmetry reductions and exact solutions to the fractional equations are presented, the compatibility of the symmetry analysis for the fractional and integer-order cases is verified. Especially, we reduce the FPDEs to the fractional ordinary differential equations (FODEs) in terms of the Erdélyi-Kober (E-K) fractional operator method, and extend the power series method for investigating exact solutions to the FPDEs.     

Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe

Generally speaking, rheological properties of materials are specified by their so-called constitutive equations. The simplest constitutive equation for a fluid is a Newtonian one, on which the classical Navier-Stokes theory is based. The mechanical behavior of many fluids is well described by this theory. However, there are many rheologically compli- cated fluids such as polymer solutions, blood and heavy oils which are inadequately de- scribed by a Newtonian constitutive equation that does …     

On the relation between the fractional Brownian motion and the fractional derivatives

The definition and simulation of fractional Brownian motion are considered from the point of view of a set of coherent fractional derivative definitions. To do it, two sets of fractional derivatives are considered: (a) the forward and backward and (b) the central derivatives, together with two representations: generalised difference and integral. It is shown that for these derivatives the corresponding autocorrelation functions have the same representations. The obtained results are used to define a fractional noise and, from it, the fractional Brownian motion. This is studied. The simulation problem is also considered.     

Fedosov Quantization of Fractional Lagrange Spaces

The main goal of this work is to perform a nonholonomic deformation (Fedosov type) quantization of fractional Lagrange–Finsler geometries. The constructions are provided for a fractional almost K?hler model encoding equivalently all data for fractional Euler–Lagrange equations with Caputo fractional derivative.     

系统非对称性及记忆性对布朗马达输运行为的影响

在对分数阶布朗马达输运现象研究的基础上,引入了描述系统势场对称性的参数(简称对称性参数),并详细分析了该参数及记忆性参数(分数阶阶数)对粒子输运状态的影响.仿真结果表明,分数阶阶数和对称性参数的共同作用会使得布朗粒子形成定向输运反向流,反向后达到最大平均流速所对应的阶数与外加驱动力频率无关联,但会随对称性参数的增加而单调递增.     

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space—time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained.     

Conformable Fractional Nikiforov-Uvarov Method

We introduce conformable fractional Nikiforov-Uvarov (NU) method by means of conformable fractional derivative which is the most natural definition in non-integer calculus. Since, NU method gives exact eigenstate solutions of Schrödinger equation (SE) for certain potentials in quantum mechanics, this method is carried into the domain of fractional calculus to obtain the solutions of fractional SE. In order to demonstrate the applicability of the conformable fractional NU method, we solve fractional SE for harmonic oscillator potential, Woods-Saxon potential, and Hulthen potential.     

Investigation of classical and fractional Bose–Einstein condensation for harmonic potential

In this study, classical and fractional Gross–Pitaevskii (GP) equations were solved for harmonic potential and repulsive interactions between the boson particles using the Homotopy Perturbation Method (HPM) to investigate the ground state dynamics of Bose–Einstein Condensation (BEC). The purpose of writing fractional GP equations is to consider the system in a more realistic manner. The memory effects of non-Markovian processes involving long-range interactions between bosons with the restriction of the ergodic hypothesis and the effect of non-Gaussian distributions of bosons in the condensation can be taken into account with time fractional and space fractional GP equations, respectively. The obtained results of the fractional GP equations differ from the results of the classical one. While the Gauss distribution describing the homogeneous, reversible and unitary system is obtained from the classical GP equation, the probability density of the solution function of fractional GP equations is non-conserved. This situation describes the inhomogeneous, irreversible and non-unitary systems.     

Kalman filters for fractional discrete-time stochastic systems along with time-delay in the observation signal

This paper investigates fractional Kalman filters when time-delay is entered in the observation signal in the discrete-time stochastic fractional order state-space representation. After investigating the common fractional Kalman filter, we try to derive a fractional Kalman filter for time-delay fractional systems. A detailed derivation is given. Fractional Kalman filters will be used to estimate recursively the states of fractional order state-space systems based on minimizing the cost function when there is a constant time delay (d) in the observation signal. The problem will be solved by converting the filtering problem to a usual d-step prediction problem for delay-free fractional systems.     

On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

By a quantum mechanical analysis of the additive rule F_α[F_β[f]] = F_α+β[f], which the fractional Fourier transformation (FrFT) F_α[f] should satisfy, we reveal that the position-momentum mutualtransformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.     

Higher-dimensional mixed fractional rotation groups as a basis for dynamic symmetries generating the spectrum of the deformed Nilsson oscillator

Based on the Riemann and Caputo definitions of the fractional derivative we use the fractional extensions of the standard rotation group SO(3) to construct a higher-dimensional representation of a fractional rotation group with mixed derivative types. An extended symmetric rotor model is derived, which predicts the sequence of magic proton and neutron numbers accurately. The ground state properties of nuclei are correctly reproduced within the framework of this model.     

Analytical Solutions,Moments, and Their Asymptotic Behaviors for the Time-Space Fractional Cable Equation

Following the fractional cable equation established in the letter [B.I. Henry, T.A.M. Langlands, and S.L.Wearne, Phys. Rev. Lett. 100(2008) 128103], we present the time-space fractional cable equation which describes the anomalous transport of electrodiffusion in nerve cells. The derivation is based on the generalized fractional Ohm's law;and the temporal memory effects and spatial-nonlocality are involved in the time-space fractional model. With the help of integral transform method we derive the analytical solutions expressed by the Green's function; the corresponding fractional moments are calculated; and their asymptotic behaviors are discussed. In addition, the explicit solutions of the considered model with two different external current injections are also presented.     

An analysis of the semi-analytic solutions of a viscous fluid with old and new definitions of fractional derivatives

In this paper we present the natural convection flow of an incompressible viscous fluid subject to Newtonian heating and constant mass diffusion using a recently developed definition of the Caputo–Fabrizio fractional derivative. Boundary layer equations in dimensionless form are obtained by means of dimensionless variables. The expressions for the temperature, concentration and velocity fields are obtained in the Laplace transformed domain. The inverse Laplace transform for the temperature, concentration and velocity field are found numerically by means of Stehfest's and Tzou's algorithms. A comparative analysis has been carried between the Caputo–Fabrizio and the Caputo fractional model obtained by Vieru (2015) through graphical illustration. At the end, we can see the impact of the flow parameters, including the new fractional parameter, on the flow which is presented graphically. As a result, the fractional viscous fluid model with the Caputo–Fabrizio fractional derivative has a higher velocity than with the Caputo.     

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries(LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation, and second, we find the related exact solutions for the derived generators. Finally,according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field Kd V equation.     

Fractional Spin of System with Chern–Simons Term Coupled to Polaron

The fractional spin of a system with Chern–Simons (CS) term coupled to a polaron at the quantum level is studied. The Faddeev–Senjanovic (FS) scheme for path-integral quantization of constrained Hamiltonian systems is applied. The quantal conserved angular momentum and the fractional spin at the quantum level of this system are presented based on the quantal Noether theorem. The fractional spin is also presented for the system with Maxwell kinetic term.     

Analytical Solutions,Moments, and Their Asymptotic Behaviors for the Time-Space Fractional Cable Equation

Following the fractional cable equation established in the letter [B.I. Henry, T.A.M. Langlands, and S.L. Wearne, Phys. Rev. Lett. 100 (2008) 128103], we present the time-space fractional cable equation which describes the anomalous transport of electrodiffusion in nerve cells. The derivation is based on the generalized fractional Ohm's law; and the temporal memory effects and spatial-nonlocality are involved in the time-space fractional model. With the help of integral transform method we derive the analytical solutions expressed by the Green's function; the corresponding fractional moments are calculated; and their asymptotic behaviors are discussed. In addition, the explicit solutions of the considered model with two different external current injections are also presented.     

Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense

Fuzzy differential equations provide a crucial tool for modeling numerous phenomena and uncertainties that potentially arise in various applications across physics, applied sciences and engineering. Reliable and effective analytical methods are necessary to obtain the required solutions, as it is very difficult to obtain accurate solutions for certain fuzzy differential equations. In this paper, certain fuzzy approximate solutions are constructed and analyzed by means of a residual power series (RPS) technique involving some class of fuzzy fractional differential equations. The considered methodology for finding the fuzzy solutions relies on converting the target equations into two fractional crisp systems in terms of ρ-cut representations. The residual power series therefore gives solutions for the converted systems by combining fractional residual functions and fractional Taylor expansions to obtain values of the coefficients of the fractional power series. To validate the efficiency and the applicability of our proposed approach we derive solutions of the fuzzy fractional initial value problem by testing two attractive applications. The compatibility of the behavior of the solutions is determined via some graphical and numerical analysis of the proposed results. Moreover, the comparative results point out that the proposed method is more accurate compared to the other existing methods. Finally, the results attained in this article emphasize that the residual power series technique is easy, efficient, and fast for predicting solutions of the uncertain models arising in real physical phenomena.