Some generalizations of comparison results for fractional differential equations

In this paper, by using the technique of upper and lower solutions together with the theory of strict and nonstrict fractional differential inequalities involving Riemann–Liouville differential operator of order q, 0<q<1, some necessary comparison results for further generalizations of several dynamical concepts are obtained. Furthermore, these results are extended to the finite systems of fractional differential equations.     

Attractivity of fractional functional differential equations

In this paper, some attractivity results for fractional functional differential equations are obtained by using the fixed point theorem. By constructing equivalent fractional integral equations, research on the attractivity of fractional functional and neutral differential equations is skillfully converted into a discussion about the existence of fixed points for equivalent fractional integral equations. Two examples are also provided to illustrate our main results.     

Numerical solution of fractional differential equations using the generalized block pulse operational matrix

The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.     

On the approximate controllability of semilinear fractional differential systems

Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of control systems governed by the semilinear fractional differential equations in Hilbert spaces. By using the semigroup theory, the fractional power theory and fixed point strategy, a new set of sufficient conditions are formulated which guarantees the approximate controllability of semilinear fractional differential systems. The results are established under the assumption that the associated linear system is approximately controllable. Further, we extend the result to study the approximate controllability of fractional systems with nonlocal conditions. An example is provided to illustrate the application of the obtained theory.     

Moment stability of fractional stochastic evolution equations with Poisson jumps

Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of fractional stochastic partial differential equations with Poisson jumps. Sufficient conditions for the existence and asymptotic stability in pth moment of mild solutions are derived by employing the Banach fixed point principle. Further, we extend the result to study the asymptotic stability of fractional systems with Poisson jumps. An example is provided to illustrate the effectiveness of the proposed results.     

分数阶微积分在图像处理中的研究综述*

综述了关于分数阶微积分理论在数字图像底层处理中的应用研究,具体包括:分数阶微积分和分数阶偏微分方程的基本理论及分数阶傅里叶变换的基本性质;分数阶微分滤波器的构造及在图像增强中的应用研究;分数阶积分滤波器的构造及在图像去噪中的应用研究;分数阶偏微分方程在图像去噪中的应用研究。最后,总结了分数阶微积分理论在图像处理中已取得的研究成果,并结合已有的基于分数阶微积分理论的图像底层处理模型,展望了分数阶微积分理论在图像处理中的应用前景。     

Application of Legendre wavelets for solving fractional differential equations

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.     

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations (FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.      

分数阶超混沌系统的线性广义同步观测器设计

首先利用分数阶的常微分动力系统的稳定性理论,通过判断线性化后平衡点的稳定不变特性、辅助以分岔图分析等数值手段,给出了新近提出的改进型超混沌L讧系统对应分数阶系统产生混沌现象的阶次参数范围;进一步,设计了一类广义线性同步观测器,该观测器的动力学行为能与原系统实现任意的线性关系的广义同步,而经典的完全同步、反相同步以及投影同步可以视为本文提出方法的特例．最后的数值仿真进一步证实了本文提出的观测器设计方案的有效性．     

A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations

This paper studies the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order q∈(1,2]. Our results are based on some standard fixed point theorems. Some illustrative examples are also discussed.     

一种自适应分数阶偏微分图像增强模型

针对传统的自适应分数阶偏微分方程图像增强算法对图像暗区纹理区域的增强不足的缺点,考虑到人眼对光感的敏感程度不同,将亮度对视觉的影响因素考虑进传统的自适应分数阶偏微分方程图像增强算法。以梯度和灰度值为参数,建立了一种新的自适应分数阶偏微分图像增强模型。该模型改善了传统算法对暗区图像增强不足的缺点,图像增强后的平均梯度提升明显,很好地改善了图像的视觉效果。实验结果说明本算法具有一定的有效性。     

Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

In this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.     

用分数阶微分提取图像边缘

文章是分数阶微分在图像处理中的尝试性应用。首先通过理论上分析得出分数阶微分可以大幅提升信号高频成分,增强信号的中频成分,非线性保留信号的甚低频。据此分析得出分数阶微分应用于图像边缘信息提取将获得高于传统基于一、二阶微分的方法的信噪比。然后由经典的分数阶微分定义出发,推导出了分数阶差分方程,构建了近似的分数阶Tiansi微分模板。最后通过图像边缘提取的实验表明：基于分数阶微分算子不仅可以有效提取图像边缘,而且比整数阶微分算子具有更高的信噪比。为拓展分数阶微分的应用领域,进行了有意义的探索。     

A unified presentation of certain families of non-Fuchsian differential equations via fractional calculus operators

In recent years, many authors demonstrated the usefulness of fractional calculus operators in the derivation of (explicit) particular solutions of a number of linear ordinary and partial differential, equations of the second and higher orders. The main object of the present paper is to show how readily some recent contributions on this subject by several workers, involving various interesting classes of non-Fuchsian differential equations (including, for example, the Fukuhara and Tricomi equations and the celebrated Bessel and Whittaker equations), can be obtained (in a unified manner) by suitably applying some general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations.     

A new method based on legendre polynomials for solution of system of fractional order partial differential equations

We study legendre polynomials in case of more than one variable and develop new operational matrices of fractional order integrations as well as fractional order differentiations. Based on these operational matrices, we develop a new sophisticated technique to solve a coupled system of fractional order partial differential equations. Our technique reduces the coupled system under consideration to a system of easily solvable algebraic equations without discretizing the system. As an application, we provide examples and numerical simulations demonstrating that the results obtained using the new technique matches well with the exact solutions of the problems. We also study error analysis graphically.     

Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part

Convergence and stability are main issues when an asymptotical method like the Homotopy Perturbation Method (HPM) has been used to solve differential equations. In this paper, convergence of the solution of fractional differential equations is maintained. Meanwhile, an effective method is suggested to select the linear part in the HPM to keep the inherent stability of fractional equations. Riccati fractional differential equations as a case study are then solved, using the Enhanced Homotopy Perturbation Method (EHPM). Current results are compared with those derived from the established Adams–Bashforth–Moulton method, in order to verify the accuracy of the EHPM. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the EHPM is powerful and efficient tool for solving nonlinear fractional differential equations.     

Existence of Optimal Mild Solutions and Controllability of Fractional Impulsive Stochastic Partial Integro‐Differential Equations with Infinite Delay

In this paper, we investigate a new class of fractional impulsive stochastic partial integro‐differential equations with infinite delay in Hilbert spaces. By using the stochastic analysis theory, fractional calculus, analytic α‐resolvent operator and the fixed point technique combined with fractional powers of closed operators, we firstly give the existence of of mild solutions and optimal mild solutions for the these equations. Next, the controllability of the controlled fractional impulsive stochastic partial integro‐differential systems with not instantaneous impulses is presented. Finally, examples are also given to illustrate our results.     

Analytical treatment of differential equations with fractional coordinate derivatives

This paper outlines a reliable strategy to use the homotopy perturbation method based on Jumarie’s derivative for solving fractional differential equations. In this framework, compact structures of fourth-order fractional diffusion-wave equations are considered as prototype examples. Moreover, convergence of the proposed approach for these types of equations is investigated. Results show that the response expressions are Mittag-Leffler stable.     

基于分数阶微分的图像增强

通过理论分析得出分数阶微分可以大幅提升信号高频成分,增强信号的中频成分、非线性保留信号的甚低频,据此得出分数阶微分应用于图像增强将使图像边缘明显突出、纹理更加清晰和图像平滑区域信息得以保留的增强图像;然后由经典的分数阶微分定义出发,推导出了分数阶差分方程,构建了近似的Tiansi微分算子.通过图像增强的实验表明:采用基于分数阶微分算子的图像增强方法,其增强图像的视觉效果明显优于传统的微分锐化(整数微分)方法.文中方法为拓展分数阶微分的应用领域进行了有意义的探索.     

The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations

In this paper, we study shifted Jacobi polynomials and develop a simple but highly accurate scheme for the numerical solution of coupled system of fractional differential equations. We derive some operational matrices of integration and differentiation of fractional order. By the application of these matrices we provide a theoretical treatment to approximate the solutions of the corresponding system. We use Matlab to perform necessary operations. The applicability of the technique is shown with some examples and the results are displayed graphically.