scholarly journals A New Definition of Fractional Derivatives Based on Truncated Left-Handed Grünwald-Letnikov Formula with0<α<1and Median Correction

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhiwu Liao
Keyword(s):  
Signal Processing ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Original Signal ◽  
Theory Analysis ◽  
Left Handed ◽  
Signal Value ◽  
Simulation Results ◽  
Definition Of ◽  
Fractional Orders

We propose a new definition of fractional derivatives based on truncated left-handed Grünwald-Letnikov formula with0<α<1and median correction. Analyzing the difficulties to choose the fractional orders and unsatisfied processing results in signal processing using fractional-order partial differential equations and related methods; we think that the nonzero values of the truncated fractional order derivatives in the smooth regions are major causes for these situations. In order to resolve the problem, the absolute values of truncated parts of the G-L formula are estimated by the median of signal values of the remainder parts, and then the truncated G-L formula is modified by replacing each of the original signal value to the differences of the signal value and the median. Since the sum of the coefficients of the G-L formula is zero, the median correction can reduce the truncated errors greatly to proximate G-L formula better. We also present some simulation results and experiments to support our theory analysis.

scholarly journals Theory Analysis of Left-Handed Grünwald-Letnikov Formula with0<α<1to Detect and Locate Singularities

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Shaoxiang Hu ◽  
Ping Liang
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Theory Analysis ◽  
Left Handed ◽  
Local Extrema ◽  
Derivative Formula ◽  
Derivatives Of

We study fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1to detect and locate singularities in theory. The widely used four types of ideal singularities are analyzed by deducing their fractional derivative formula. The local extrema of fractional derivatives are used to locate the singularities. Theory analysis indicates that fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1can detect and locate four types of ideal singularities correctly, which shows better performance than classical 1-order derivatives in theory.

HYPERCHAOTIC BEHAVIOR IN ARBITRARY-DIMENSIONAL FRACTIONAL-ORDER QUANTUM CELLULAR NEURAL NETWORK MODEL

2013 ◽  
Vol 23 (03) ◽  
pp. 1350044 ◽  
Author(s):  
LING LIU ◽  
CHONGXIN LIU ◽  
DELIANG LIANG
Keyword(s):  
Neural Network ◽  
Signal Processing ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Cellular Neural Network ◽  
Cell Models ◽  
Fractional Quantum ◽  
Order Form ◽  
Simulation Results ◽  
Parallel Signal Processing

In this paper, an arbitrary-dimensional quantum cellular neural network (QCNN) model and its fractional-order form are presented by using the polarization of quantum-dot cell and fractional derivatives. Two classes of fractional-order QCNN equations, either two-cell or three-cell models with different value of fractional order α are taken into consideration in detail. In particular, more complex and abundant fractional-order hyperchaotic behaviors can be observed by these two examples. Thus, the proposed fractional-order arbitrary-dimensional QCNN model can have an effective noninteger dimension and can generate rich hyperchaotic dynamics by nth cell. Numerical analysis and simulation results are provided to show the effectiveness of the proposed approach. This study provides valuable information about nth cell fractional-order QCNNs for further application in high-parallel signal processing and fractional quantum chaotic generators.

scholarly journals Modelowanie układów sterowania z użyciem pochodnych ułamkowych

2020 ◽  
pp. 33-50
Author(s):  
Grzegorz Dec ◽  
Keyword(s):  
Experimental Data ◽  
System Theory ◽  
Control System ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
System Control ◽  
Current State ◽  
Function Definition ◽  
Definition Of ◽  
State Of Art

In the paper is presented review of some approaches corelated with subject of using fractional derivatives in control system theory. Popular algorithms used in the industry are presented, along with relating designing methodology. Using of fractional derivatives calculations is relatively new concept, but constantly getting increasing interest. Deliberation in recent years indicate that many scientific problems like thermodynamic or biology problems can be well considered and modeled by fractional order derivatives. On the market there is available tools that support a processes of identification and regulators designing, based on experimental data. One of such tools are toolbox CRONE for MATLAB, which contains three modules: mathematical, identifying, system control designing. That toolbox allows implementation of CRONE regulators with different level of complexity. Other tool is FOMCON, which also is a toolbox for MATLAB and it is based on already existed toolbox FOTF. FOMCON allows to identifying of control system and PIλDµ regulator designing. This article is aiming to present current state of art, discussion about existing tools and concepts correlated with fractional order derivatives and their usage in control system theory, like: gamma function, definition of fractional derivative, Laplace transform and basics of control system theory.

scholarly journals Wigner-Ville Distribution Associated with the Linear Canonical Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Rui-Feng Bai ◽  
Bing-Zhao Li ◽  
Qi-Yuan Cheng
Keyword(s):  
Signal Processing ◽  
Signal Detection ◽  
Transform Domain ◽  
Linear Canonical Transform ◽  
Nonstationary Signal ◽  
Linear Frequency ◽  
Simulation Results ◽  
Definition Of ◽  
Linear Frequency Modulated Signal

The linear canonical transform is shown to be one of the most powerful tools for nonstationary signal processing. Based on the properties of the linear canonical transform and the classical Wigner-Ville transform, this paper investigates the Wigner-Ville distribution in the linear canonical transform domain. Firstly, unlike the classical Wigner-Ville transform, a new definition of Wigner-Ville distribution associated with the linear canonical transform is given. Then, the main properties of the newly defined Wigner-Ville transform are investigated in detail. Finally, the applications of the newly defined Wigner-Ville transform in the linear-frequency-modulated signal detection are proposed, and the simulation results are also given to verify the derived theory.

scholarly journals Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 125 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Dumitru Baleanu
Keyword(s):  
Fractional Order ◽  
Traveling Wave ◽  
Fractional Derivatives ◽  
Traveling Wave Solution ◽  
Order System ◽  
Transform Method ◽  
Homotopy Analysis ◽  
Definition Of ◽  
Analytical Approaches ◽  
Natural Decomposition

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science.

scholarly journals The influence of the Lorenz system fractionality on its recurrensivity

2019 ◽  
Vol 252 ◽  
pp. 02006
Author(s):  
Magdalena Gregorczyk ◽  
Andrzej Rysak
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
System Dynamics ◽  
Phase Diagrams ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Lorenz System ◽  
Recurrence Analysis ◽  
The Lorenz System ◽  
Fractional Orders

In this work, we investigate the recurrensivity of the Lorenz system with fractional order of derivatives occurring in its all three differential equations. Several solutions of the system for varying fractional orders of individual derivatives were calculated, which was followed by an analysis of changes in the selected recurrence quantifiers occurring due to modifications of the fractional orders {q1, q2, q3}. The results of the recurrence analysis were referred to the time series plots, phase diagrams and FFT spectra. The obtained results were finally used to examine the influence of fractional derivatives on the chaos - periodicity transition of the system dynamics.

scholarly journals A discrete time method to the first variation of fractional order variational functionals

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Shakoor Pooseh ◽  
Ricardo Almeida ◽  
Delfim Torres
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Effective Solution ◽  
First Variation ◽  
First Order ◽  
Variational Functional ◽  
Discrete Nature ◽  
The First Variation ◽  
Definition Of

AbstractThe fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grünwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations.

Solutions of General Fractional-Order Differential Equations by Using the Spectral Tau Method

2021 ◽  
Vol 6 (1) ◽  
pp. 7
Author(s):  
Hari Mohan Srivastava ◽  
Daba Meshesha Gusu ◽  
Pshtiwan Othman Mohammed ◽  
Gidisa Wedajo ◽  
Kamsing Nonlaopon ◽  
...  
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Convergent Series ◽  
Initial Condition ◽  
Tau Method ◽  
Series Solutions ◽  
Fractional Order Differential Equations ◽  
Functional Argument ◽  
Fractional Orders

Here, in this article, we investigate the solution of a general family of fractional-order differential equations by using the spectral Tau method in the sense of Liouville–Caputo type fractional derivatives with a linear functional argument. We use the Chebyshev polynomials of the second kind to develop a recurrence relation subjected to a certain initial condition. The behavior of the approximate series solutions are tabulated and plotted at different values of the fractional orders ν and α. The method provides an efficient convergent series solution form with easily computable coefficients. The obtained results show that the method is remarkably effective and convenient in finding solutions of fractional-order differential equations.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

scholarly journals Fractional-Order Colour Image Processing

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 457
Author(s):  
Manuel Henriques ◽  
Duarte Valério ◽  
Paulo Gordo ◽  
Rui Melicio
Keyword(s):  
Image Processing ◽  
Edge Detection ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Grey Scale ◽  
Colour Image ◽  
Colour Image Processing ◽  
Image Processing Algorithms ◽  
Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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