Approximation of Laplace transform of fractional derivatives via Clenshaw–Curtis integration

In this paper, we approximate the Laplace transform of fractional derivatives via Clenshaw–Curtis integration. The idea of applying Chebyshev polynomial to the numerical computation of integrals is extended to Laplace transform of fractional derivatives. The numerical stability of forward recurrence relations is considered, which depends on the asymptotic behaviour of the coefficients. Error estimation for the Laplace approximation of the fractional derivatives is also considered. Finally, from the numerical examples, the method seems to be promising for approximation of the Laplace transform of fractional derivative.     

Estimation of parameters in fractional subdiffusion equations by the time integral characteristics method

This paper presents an extension to the time integral characteristics method for estimation of parameters in fractional subdiffusion equations containing Riemann-Liouville and Caputo fractional time derivatives. The explicit representations of the fractional diffusion coefficient and order of fractional differentiation via a Laplace transform of the concentration field are obtained. A technique of optimal Laplace parameter determination by minimization of relative errors bounds is described. The effectivity of the proposed approach is illustrated by numerical example.     

基于局部特征的分数阶微分图像增强方法

一般情况下分数阶微分模板一经确定,再用其进行滤波时并不随图像的局部信息而变化,它不具有灵活性。针对分数阶微分模板滤波的这种局限性,提出了一种基于局部特征的分数阶微分图像增强的方法。在3×3对称分数阶微分模板的基础上找出与拉普拉斯模板的关系,从而得到加权的拉普拉斯模板表示的分数阶微分模板;根据图像的局部均值与标准差的关系对加权的拉普拉斯模板进一步改进,得到基于局部特征的分数阶微分图像增强的方法,它使分数阶图像增强模板能够根据局部特征灵活地进行滤波。将其与其他的图像增强算法比较,实验证明基于局部特征的分数阶微分图像增强算法能获得更好效果。     

Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping methods, we develop the Laplace transform methods for solving the free-boundary fractional diffusion equations. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace discretizing the temporal variable. The hyperbola contour integral method is exploited to restore the option values. Meanwhile, the coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation by the fast Laplace transform method is guaranteed. The numerical results confirm that the proposed method outperforms the full finite difference methods in regard to the accuracy and complexity.     

用零极点动态控制波形图显示的Java小程序实现方法

用Java小程序实现了通过拉普拉斯逆变换和z逆变换中s域或z域的零极点动态控制波形图显示的过程。为了增强人机界面的交互性和稳定性,程序采用了Java语言中的鼠标事件来捕获零极点坐标从而动态控制波形图显示,并且也应用了双缓冲技术来消除擦旧画新过程所引起的画面闪烁。实际应用表明,用户通过此程序可以更加方便直观地理解诸如系统稳定性等一系列问题,达到传统教学系统所没有的效果。     

Generalization of generalized orthogonal polynomial operational matrices for fractional and operational calculus

A method is given for the approximation of generalized orthogonal polynomials (GOP) to solve the problems of fractional and operational calculus. A more rigorous derivation for the generalized orthogonal polynomial operational matrices is proposed. The Riemann-Liouville fractional integral for repeated fractional (and operational) integration is integrated exactly, then expanded in generalized orthogonal polynomials to yield the generalized orthogonal polynomial operational matrices. The generalized orthogonal polynomial operational matrices perform as s^α(α ≥ αε R) in the Laplace domain and as fractional (and operational) integrators in the time domain. Using these results, inversions of the Laplace transforms of irrational and rational transfer functions are solved in a simple way. Very accurate results are obtained.     

分数阶混沌系统同步及其保密通信

针对参数未知的分数阶Chen混沌系统,研究其同结构同步以及与分数阶Lü混沌系统的异结构同步问题。利用分数阶系统稳定性理论和拉普拉斯变换理论,设计并证明了系统的反馈控制器,给出了一种分数阶混沌保密通信系统。运用分数阶微积分的预估——校正算法进行数值仿真,验证了所提出方法的有效性。     

Closed form of the Solutions of some Boundary-Value Problems for Anomalous Diffusion Equation with Hilfer’s Generalized Derivative

The concept of double-order fractional derivative generalizing the well-known Hilfer’s derivative is introduced. The formula is given for the Laplace transform of double-order fractional derivative, which is used to solve the Cauchy-type problem for equations of fractional order with this derivative. The closed solutions to some boundary-value problems for the equation of anomalous diffusion with double-order fractional derivative in time are obtained.     

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。     

分数阶线性系统的内部和外部稳定性研究

介绍了分数阶线性定常系统的状态方程描述和传递函数描述．运用拉普拉斯变换和留数定理，给出并证明了分数阶线性定常系统的内部和外部稳定性条件，并讨论了其相互关系．以一个粘弹性系统的实例验证了上述方法的正确性．     

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

The paper focuses on the numerical approximation of discrete fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed model of discrete fractional order system is introduced. Then, resorting the nabla discrete Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology.     

Riesz fractional order derivative in Fractional Fourier Transform domain: An insight

This paper presents a novel closed-form analytical expression for Riesz fractional order derivative in the Fractional Fourier domain. The expression is obtained in the terms of higher transcendental functions such as Parabolic Cylinder Function as well as Confluent Hypergeometric Function. The presented work is analyzed in the discrete domain by using the properties of Discrete Fractional Fourier Transform (DFrFT). The proposed algorithm is capable of preserving the texture and edge information without any phase distortion. The design example discussed in the paper shows the efficacy of the proposed algorithm for a signal with high frequency chirp noise. The design flexibility of the proposed approach is confirmed due to the fact that it provides an optimal value of performance metrics such as Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) corresponding to the variation of the fractional order of Riesz derivative and fractional parameter in the rotation angle of Fractional Fourier Transform (FrFT). The proposed algorithm provides better results in terms of minimum RMSE of 0.115136 and MAE of 0.094223 for the optimal fractional order of 0.43 at a rotation angle of 0.45π.     

Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Transform techniques

An analytical solution for the transient two-dimensional atmospheric pollutant dispersion problem is presented. The approach used in this problem utilizes the double GITT (Generalized Integral Transform Technique), the Laplace Transform and the matrix diagonalization. Furthermore, mathematical filters are used due to the existence of non-homogeneous boundary conditions. The results we obtained are compared with experimental data for short range downwind dispersion utilizing two well-known experimental dispersion datasets (Copenhagen and Prairie Grass). It is shown that the present analytical approach give good results for downwind concentration except for receptors very close to the release points were any Eulerian approach based on K-closure is known to fail.     

一种基于分数阶Fourier域的数字水印

提出了一种分数阶Fourier域的水印嵌入算法。将一复伪随机序列作为水印信息嵌入到图像的分数阶Fourier域中。分数阶Fourier变换的变换角度(α,β)为水印增加了两个自由度,增强了水印的安全性。仿真结果验证了该算法的有效性。     

Necessary and Sufficient Conditions for Consensus of Delayed Fractional‐order Systems

In this paper, we study the consensus problem of fractional‐order systems with input delays. Using Laplace transform method, the stability of the fractional‐order systems is first discussed in the frequency domain. Based on the generalized Nyquist stability criterion, a necessary and sufficient condition is further derived to ensure the consensus of fractional‐order systems with identical input delays over directed interaction topology. Furthermore, when the interaction topology is undirected, the consensus condition of fractional‐order systems with heterogeneous input delays is explicitly given. Finally, some illustrative examples are presented to show the effectiveness and advantages of the theoretical results.     

Delay-induced bifurcation in a tri-neuron fractional neural network

This paper investigates the issue of stability and bifurcation for a delayed fractional neural network with three neurons by applying the sum of time delays as the bifurcation parameter. Based on fractional Laplace transform and the method of stability switches, some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. The analysis indicates that time delay can effectively enhance the stability of fractional neural networks. In addition, it is found that the stability interval can be varied by regulating the fractional order if all the parameters are fixed including time delay. Finally, numerical examples are presented to validate the derived theoretical results.     

二维离散分数余弦变换的改进形式

将一维离散余弦变换的变换核扩展到二维分数形式,得到Pei形式的二维离散分数余弦变换。通过整数阶余弦变换的线性叠加构造一种改进形式的二维离散分数余弦变换,并基于特征值和特征向量理论,分析2种离散分数余弦变换的周期关系。数值仿真结果表明,2种形式可以达到相同的变换结果,适用于图像编码、数字水印等领域。     

Generalized fractional magneto-thermo-viscoelasticity

The new model of the equations of the linear theory of magneto-thermo-viscoelasticity with two relaxation times and fractional heat transfer involving fractional relaxation operator is given. The resulting formulation is applied to thermal shock problems for a perfect electrically conducting half-space in the presence of a transverse magnetic field. Laplace transform techniques are used. Some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity with two relaxation times are established. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The effects of the fractional relaxation operator on viscoelastic material like poly (methyl methacrylate) (Perspex) are discussed.

     

基于分数阶微分的尺度不变特征变换图像匹配算法

利用分数阶微积分运算处理图像信息,有利于强化和提取图像的纹理细节,使图像得到增强,更有利于对图像特征的提取。为了提高图像匹配的正确性,用基于分数阶微积分图像处理方法,提出了改进的尺度不变特征变换(SIFT)匹配算法,将高斯滤波和分数阶微分滤波相结合,用分数阶微分对图像特征进行强化,检测出更加稳定的尺度空间极值点,然后筛选出更多和更准确的匹配特征点,最后进行图像匹配。实验表明,在SIFT中引入分数阶微积分的应用,能够得到更多的特征关键点,提高图像匹配的正确性。