基于分数傅里叶变换的分数相关峰值特性

本文通过理论分析和计算机仿真研究了分数傅里叶变换的级次对分数相关峰值特性的影响并优化了分数相关的级次。结果表明分数相关输出在旁辩和峰值宽度方面与传统相关相比有了较大的改善,因而可以提高目标探测的灵敏性。     

The application of fractional derivatives in stochastic models driven by fractional Brownian motion

In this paper, in order to establish connection between fractional derivative and fractional Brownian motion (FBM), we first prove the validity of the fractional Taylor formula proposed by Guy Jumarie. Then, by using the properties of this Taylor formula, we derive a fractional Itô formula for H∈[1/2,1), which coincides in form with the one proposed by Duncan for some special cases, whose formula is based on the Wick Product. Lastly, we apply this fractional Itô formula to the option pricing problem when the underlying of the option contract is supposed to be driven by a geometric fractional Brownian motion. The case that the drift, volatility and risk-free interest rate are all dependent on t is also discussed.     

Fractional variational iteration method and its application

Fractional differential equations have been investigated by variational iteration method. However, the previous works avoid the term of fractional derivative and handle them as a restricted variation. We propose herein a fractional variational iteration method with modified Riemann Liouville derivative which is more efficient to solve the fractional differential equations.     

Solutions of Fractional Partial Differential Equations of Quantum Mechanics

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox's $H$-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177-190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.     

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.     

Optical image encryption with redefined fractional Hartley transform

A new method for optical image encryption is introduced on the basis of two-dimensional (2-D) generalization of 1-D fractional Hartley transform that has been redefined recently in search of its inverse transform. We encrypt the image by two fractional orders and random phase codes. It has an advantage over Hartley transform, for its fractional orders can also be used as additional keys, and that, of course, strengthens image security. Only when all of these keys are correct, can the image be well decrypted. The optical realization is then proposed and computer simulations are also performed to confirm the possibility of the proposed method.     

Solving fractional diffusion and wave equations by modified homotopy perturbation method

This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and Odibat to obtaining solutions of linear and nonlinear fractional diffusion and wave equations. The fractional derivative is described in the Caputo sense. Some illustrative examples are given, revealing the effectiveness and convenience of the method.     

Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology

In this study, the modified Kudryashov method is used to construct new exact solutions for some conformable fractional differential equations. By implementing the conformable fractional derivative and compatible fractional complex transforms, the fractional generalized reaction duffing (RD) model equation, the fractional biological population model and the fractional diffusion reaction (DR) equation with quadratic and cubic nonlinearity are discussed. As an outcome, some new exact solutions are formally established. All solutions have been verified back into its corresponding equation with the aid of maple package program. We assure that the employed method is simple and robust for the estimation of the new exact solutions, and practically capable for reducing the size of computational work for solving a various class of fractional differential equations arising in applied mathematics, mathematical physics and biology.     

Relationships between power-law long-range interactions and fractional mechanics

We investigate the relationships between models of power-law long-range interactions and mechanics based on fractional derivatives. We present the fractional Lagrangian density which gives the Euler–Lagrange equation that serves as the equation of motion for fractional-power-law long-range interactions. We derive this equation by the fractional variational method. In addition, we derive a Noether-like current from the fractional Lagrangian density.     

Fractional dynamics of systems with long-range space interaction and temporal memory

Field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations. As another example, dynamical equations for particle chains with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.     

Fractional Zero Curvature Equation and Generalized Hamiltonian Structure of Soliton Equation Hierarchy

We presented the fractional zero curvature equation and generalized Hamiltonian structure by using of the differential forms of fractional orders. Example of the fractional AKNS soliton equation hierarchy and its Hamiltonian system are obtained.     

Image encryption with fractional wavelet packet method

We introduce a new method called fractional wavelet packet transform to encrypt images in this paper, in which fractional orders and wavelet packet filter are its two series of keys. Fractional orders are additional keys in this method compared to wavelet packet encryptions. Selected image encryption is also proposed in this paper, and it is quite more flexible and effective than wavelet, fractional wavelet or wavelet packet encryptions. The possible optical implementation and digital computation are proposed. Computer simulations prove its feasibility.     

A numerical method for fractional Schrödinger equation of Lennard-Jones potential

Depending on fractional analysis, we find a numerical algorithm to solve the time-independent fractional Schrödinger equation in case of Lennard-Jones potential in one dimension. We apply the algorithm for multiple values of the fractional parameter of the space-dependent fractional Schrödinger equation and multiple values of the system's energy to find the wave function and the probability in these cases.     

Theory of four-dimensional fractional quantum Hall states

We propose a pseudo-potential Hamiltonian for the Zhang-Hu’s generalized fractional quantum Hall states to be the exact and unique ground states. Analogously to Laughlin’s quasi-hole (quasi-particle), the excitations in the generalized fractional quantum Hall states are extended objects. They are vortex-like excitations with fractional charges +(−)1/m³ in the total configuration space CP³. The density correlation function of the Zhang-Hu states indicates that they are incompressible liquid.     

Fokker-Planck equation with fractional coordinate derivatives

Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1<α<2.     

Uniform of four fractional-order nonlinear feedback synchronizations

This paper designs four fractional order nonlinear feedback synchronizations with the simple configuration, followed with their uniform. The closed system's stability is proved based on the fractional order stability theory. Resorted to the fractional order unified chaotic system, it is illustrated that the uniform includes the active, ordinary, dislocated, speed nonlinear feedback synchronizations and their mixed formulations. Numerical simulations show the effectiveness of the proposed methods.     

A generalized Tu formula and Hamiltonian structures of fractional AKNS hierarchy

In this Letter, a generalized Tu formula is firstly presented to construct Hamiltonian structures of fractional soliton equations. The obtained results can be reduced to the classical Hamiltonian hierarchy of AKNS in ordinary calculus.     

Fractional generalized Langevin equation approach to single-file diffusion

Fractional generalized Langevin equation with external force is used to model single-file diffusion. It is found that for external force that varies with power law the solution for such a fractional Langevin equation gives the correct short and long time behavior for the mean square displacement of single-file diffusion when appropriate choice of parameters associated with fractional generalized Langevin equation are used. By considering some special cases of the fractional generalized Langevin equation, a new class of closed analytic expressions for the mean square displacement of single-file diffusion can be obtained. The effective Fokker-Planck equation associated with single-file diffusion is briefly considered.     

基于随机分数傅里叶变换的双图像加密算法

利用光学随机分数傅里叶变换设计了一种双图像加密算法,并给出了相应的光学实现.加密算法中,将两幅原始图像分别作为加密系统输入复函数的振幅和位相分布函数,利用随机分数傅里叶变换进行加密,所得复函数的振幅即为加密图像,而位相部分是变换的输出相位,随机位相作为加密算法的密码.在数值模拟中,二值文本图像和灰度图像分别被作为原始图像用于加密结果分析和加密安全测试,结果表明该加密算法具有很好的安全性.     

COS光束的零场截取性质

运用零场截取技术由COS光束得到COS芯(中央斑),数值模拟了COS芯通过不同的分数高斯光阑和分数高斯相位光阑后,在远场处的横向光强分布特性.结果表明,只要通过选取合适的光阑参量,COS芯传输后,在远场看来具有无旁斑的特性.