A tau approach for solution of the space fractional diffusion equation

Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.     

分数阶非线性Duffing振子方程的特性研究

将Caputo分数阶微分算子引入到非线性的Duffing振子方程中,运用同伦扰动变换法--一种同伦扰动法和Laplace变换相结合的方法来求解分数阶的非线性方程,借助Mathematica软件的符号计算功能得到了分数阶非线性Duffing振子方程的近似解,研究了振子运动过程与分数阶导数之间的关系。     

Research progress on discretization of fractional Fourier transform

As the fractional Fourier transform has attracted a considerable amount of attention in the area of optics and signal processing, the discretization of the fractional Fourier transform becomes vital for the application of the fractional Fourier transform. Since the discretization of the fractional Fourier transform cannot be obtained by directly sampling in time domain and the fractional Fourier domain, the discretization of the fractional Fourier transform has been investigated recently. A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented in this paper. The discretizations include sampling in the fractional Fourier domain, discrete-time fractional Fourier transform, fractional Fourier series, discrete fractional Fourier transform （including 3 main types： linear combination-type; sampling-type; and eigen decomposition-type）, and other discrete fractional signal transform. It is hoped to offer a doorstep for the readers who are interested in the fractional Fourier transform.     

一种基于正弦变换的三维泊松方程并行求解算法

泊松方程的数值解法在许多物理或者工程问题上得到广泛应用,但是由于大部分三维泊松方程的离散化格式不具有明显的并行性,实际中使用整体迭代的思想,这使得计算效率和稳定性受到了限制。摒弃了传统数值解法中整体迭代的思想,结合离散正弦变换理论(DST),基于27点四阶差分格式,将三维泊松方程求解算法在算法级进行修改和并行优化,把整个求解问题转化成多个独立的问题进行求解,稳定性和并行性能得到大幅提升。对于确定的离散化形式,可以使用同一套参数解决不同的泊松方程,大大提高了编程效率。基于共享存储并行模型实现了该算法,实验结果显示,对于给出的实例,新算法具有较好的加速效果,计算结果精度误差约为10e-5,在可接受范围内,并且计算精度随着维数的升高具有一定提升。     

一种新型分数阶小波变换及其应用

小波变换和分数Fourier变换是应用非常广泛的信号处理工具.但是,小波变换仅局限于时频域分析信号;分数Fourier变换虽突破了时频域局限能够在分数域分析信号,却无法表征信号局部特征.为此,提出了一种新型分数阶小波变换,该变换不但继承了小波变换多分辨分析的优点,而且具有分数Fourier变换分数域表征功能.与现有分数阶小波变换相比,新型分数阶小波变换可以实现对信号在时间-分数频域的多分辨分析.此外,该变换具有物理意义明确和计算复杂度低的优点,更有利于满足实际应用需求.最后,通过仿真实验验证了所提理论的有效性.     

A novel fractional wavelet transform and its applications

The wavelet transform (WT) and the fractional Fourier transform (FRFT) are powerful tools for many applications in the field of signal processing.However,the signal analysis capability of the former is limited in the time-frequency plane.Although the latter has overcome such limitation and can provide signal representations in the fractional domain,it fails in obtaining local structures of the signal.In this paper,a novel fractional wavelet transform (FRWT) is proposed in order to rectify the limitations of the WT and the FRFT.The proposed transform not only inherits the advantages of multiresolution analysis of the WT,but also has the capability of signal representations in the fractional domain which is similar to the FRFT.Compared with the existing FRWT,the novel FRWT can offer signal representations in the time-fractional-frequency plane.Besides,it has explicit physical interpretation,low computational complexity and usefulness for practical applications.The validity of the theoretical derivations is demonstrated via simulations.     

A wavelet approach for the multi-term time fractional diffusion-wave equation

In this paper, a Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is established for solving the multi-term time fractional diffusion-wave equation. To do this, a new operational matrix of fractional integration for the SKCWs must be derived and in order to improve the computational efficiency, the hat functions are proposed to create a general procedure for constructing this matrix. Implementation of these wavelet basis functions and their operational matrix of fractional integration simplifies the problem under consideration to a system of linear algebraic equations, which greatly decreases the computational cost for finding an approximate solution. The main privilege of the proposed method is adjusting the initial and boundary conditions in the final system automatically. Theoretical error and convergence analysis of the SKCWs expansion approve the reliability of the approach. Also, numerical investigation reveals the applicability and accuracy of the presented method.     

应用分数阶傅里叶变换的形状描述方法研究

傅里叶描述子是一种经典的形状描述方法。作为傅里叶变换的推广形式,分数阶傅里叶变换在数字信号处理工程领域已有相当广泛的应用,但在形状分析领域还很少有研究工作的报道。首次研究了基于分数阶傅里叶变换的形状描述方法,比较了不同阶数下的分数阶傅里叶描述子在图像检索中的性能。通过在MPEG-7的标准图像测试集的图像检索实验,得出：阶数ρ为0.1时,分数阶傅里叶描述子的检索效果最差,随ρ=0.1的增长,检索性能总体呈上升趋势,当ρ=0.5变化到1.0时,检索性能最高。同时,与Zernike矩进行比较：当阶数为0.1时,分数阶傅里叶描述子的检索性能较差;而阶数为0.5、1.0时分数阶傅里叶描述子的检索性能均较好。     

对称分数B样条小波的PCA图像融合

有良好逼近能力的对称分数B样条小波,在刻画图像纹理方面优于传统小波,为图像融合提供了有利条件。将其与PCA（Principal Component Analysis）变换相结合之后对高分辨率全色图像和低分辨率多光谱图像进行融合,提出了一种新的图像融合算法。对两幅源图像应用PCA变换,得到的两个第一主分量分别进行对称分数B样条小波变换,再对产生的两组高、低频小波系数采取不同的规则进行融合,生成两组新的高、低频系数,对其进行小波反变换得到新的第一主分量,与多光谱图像的其他主分量进行PCA反变换,得到最终的融合图像。实验结果表明,该方法使融合图像既提高了分辨率又保留了丰富的光谱信息。     

A generalized fractional KN equation hierarchy and its fractional Hamiltonian structure

A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders. We construct the generalized fractional trace identity through the Riemann-Liouville fractional derivative. An example of the fractional KN soliton equation hierarchy and Hamiltonian structure is presented, which is a new integrable hierarchy and possesses Hamiltonian structure.     

Spectral regularization method for the time fractional inverse advection-dispersion equation

In this paper, we consider the time fractional inverse advection-dispersion problem (TFIADP) in a quarter plane. The solute concentration and dispersion flux are sought from a measured concentration history at a fixed location inside the body. Such problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α(0 < α < 1). We show that the TFIADP is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

基于分数阶Fourier变换的数字图像加密算法研究*

基于分数阶Fourier变换和混沌,提出了一种数字图像加密方法。具体算法为：先对图像进行混沌置乱,再进行X方向的离散分数阶Fourier变换;然后在分数阶Fourier域内作混沌置乱,再进行Y方向的离散分数阶Fourier变换;最后将加密图像的实部与虚部映射到RGB,形成可传输的彩色加密图像。实验结果表明,该加密算法具有很好的安全性,在信息安全领域有较好的应用前景和研究价值。     

A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order

We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs.     

基于快速分数阶傅氏变换的DDoS攻击检测

针对传统检测方法存在精度低、训练复杂度高、适应性差的问题,提出了基于快速分数阶Fourier变换估计Hurst指数的DDoS攻击检测方法。利用DDoS攻击对网络流量自相似性的影响,通过监测Hurst指数变化阈值判断是否存在DDoS攻击。在DARPA2000数据集和不同强度TFN2K攻击流量数据集上进行了DDoS攻击检测实验,实验结果表明,基于FFrFT的DDoS攻击检测方法有效,相比于常用的小波方法,该方法计算复杂度低,实现简单,Hurst指数估计精度更高,能够检测强度较弱的DDoS攻击,可有效降低漏报、误报率。     

量子混沌和分数阶Fourier变换的图像加密算法

针对传统的自然混沌系统安全性低的问题,提出了量子混沌和分数阶Fourier变换的图像加密算法。通过引入量子Logistic混沌映射,解决了Logistic映射存在的周期窗口、伪随机和非周期性不好等缺陷,还改善了计算机进行浮点数运算丢失精度的问题。同时将混沌系统和分数阶Fourier变换相结合,实现了介于空间域和频域的分数域置乱,克服了传统一些方法只在单一域变换和单纯使用某一种方案而导致参数变量少,系统结构简单,直方图不均匀等缺点。实验和仿真结果表明,该算法具有密钥空间大,计算复杂度低,敏感性强等优点,能够有效地抵御统计分析攻击。     

A fusion estimation method based on fractional Fourier transform

Image denoising methods have different denoising performance in both spatial and transform domains, and each method has its relative advantages and inherent shortcomings compared with other methods. A very intuitive idea is to find that an effective fusion method that can combine with the advantages of different denoising methods. In this paper, we propose a novel fusion method based on the fractional Fourier transform and apply it to image denoising problem. Our method is mainly divided into three steps: Firstly, a pre-estimation is made by any two denoising method separately in the spatial domain. Secondly, using these two estimated results as well as their Fourier transform, twice Fourier transform and three times Fourier transform, we obtain a fused result in the fractional Fourier transform domain. Thirdly, the inverse fractional Fourier transform and the modulus operation are used to obtain the final fusion result. Obviously, this approach is the fusion method in four different domains. Experimental results on benchmark test images demonstrate that the proposed method outperforms state-of-the-art stand-alone methods as: BM3D, DDID, MLP, EPLL and also superior to the fusion methods such as classic wavelet fusion method, PCA fusion method and the state-of-the-art CIEM fusion method in terms of quantity value such as the peak signal to noise ratio (PSNR), the structural similarity (SSIM), and visual quality.     

The fractional Fourier transform: theory, implementation and error analysis

The fractional Fourier transform is a time–frequency distribution and an extension of the classical Fourier transform. There are several known applications of the fractional Fourier transform in the areas of signal processing, especially in signal restoration and noise removal. This paper provides an introduction to the fractional Fourier transform and its applications. These applications demand the implementation of the discrete fractional Fourier transform on a digital signal processor (DSP). The details of the implementation of the discrete fractional Fourier transform on ADSP-2192 are provided. The effect of finite register length on implementation of discrete fractional Fourier transform matrix is discussed in some detail. This is followed by the details of the implementation and a theoretical model for the fixed-point errors involved in the implementation of this algorithm. It is hoped that this implementation and fixed-point error analysis will lead to a better understanding of the issues involved in finite register length implementation of the discrete fractional Fourier transform and will help the signal processing community make better use of the transform.     

MIMO-OFDM system based on fractional Fourier transform and selecting algorithm for optimal order

In the rapidly time-varying channel environment, the performance of traditional MIMO-OFDM system is deteriorated due to the intercarrier interference. In this paper, a novel MIMO-OFDM system is proposed, in which the modulation and de- modulation of the symbols are implemented by the fractional Fourier transform instead of traditional Fourier transform. Through selecting the optimal order of the fractional Fourier transform, the modulated signals can match the time-varying channel characteristics, which results in a mitigation of the intercarrier interference. Furthermore, an algorithm is presented for selecting the optimal order of fractional Fourier transform, and the impact of system parameters on the optimal order is analyzed. Simulation results show that the proposed system can concentrate the power of desired signal effectively and improve the performance over rapidly time-varying channels with respect to the traditional MIMO-OFDM system.     

Chebyshev polynomial and heat conduction equation in cylindrical geometry

The solutions of the unsteady heat conduction equations in cylindrical geometry in one and two dimensions are obtained using the Chebyshev polynomial expansions in the spatial domain. Equations are discretized in the time domain using the trapezoidal rule. The resulting differential equations are reduced to backward recurrence relations for the coefficients occurring in the Chebyshev polynomial expansions, which are then solved using the Tau method. It is shown that the Chebyshev polynomial solutions produce results to the machine-precision accuracy in the spatial domain using only a modest number of terms, and are, therefore, excellent alternatives to the other techniques used.