scholarly journals Fractional calculus in image processing: a review

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Qi Yang ◽  
Dali Chen ◽  
Tiebiao Zhao ◽  
YangQuan Chen
Keyword(s):  
Image Processing ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Free Parameter ◽  
Science And Engineering ◽  
Fractional Systems ◽  
Integer Order

AbstractOver the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional systems. Due to the extra free parameter order

scholarly journals Razumikhin Stability Theorem for Fractional Systems with Delay

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
D. Baleanu ◽  
S. J. Sadati ◽  
R. Ghaderi ◽  
A. Ranjbar ◽  
T. Abdeljawad (Maraaba) ◽  
...  
Keyword(s):  
Time Delay ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Stability Theorem ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Science And Engineering ◽  
Fractional Systems ◽  
Razumikhin Theorem ◽  
The Stability

Fractional calculus techniques and methods started to be applied successfully during the last decades in several fields of science and engineering. In this paper we studied the stability of fractional-order nonlinear time-delay systems for Riemann-Liouville and Caputo derivatives and we extended Razumikhin theorem for the fractional nonlinear time-delay systems.

scholarly journals Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Yi-Fei Pu ◽  
Ji-Liu Zhou ◽  
Patrick Siarry ◽  
Ni Zhang ◽  
Yi-Guang Liu
Keyword(s):  
Differential Equation ◽  
Image Processing ◽  
Partial Differential Equation ◽  
Fractional Order ◽  
Differential Calculus ◽  
Order Partial Differential Equation ◽  
Texture Image ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Integer Order

The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.

Fractional Meissner–Ochsenfeld effect in superconductors

2019 ◽  
Vol 33 (26) ◽  
pp. 1950316 ◽  
Author(s):  
J. F. Gómez-Aguilar
Keyword(s):  
Magnetic Field ◽  
Critical Temperature ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Mathematical Tool ◽  
Conformable Derivative ◽  
Science And Engineering ◽  
Interest In Science ◽  
London Equation ◽  
Physical Phenomena

Fractional calculus (FC) is a valuable tool in the modeling of many phenomena, and it has become a topic of great interest in science and engineering. This mathematical tool has proved its efficiency in modeling the intermediate anomalous behaviors observed in different physical phenomena. The Meissner–Ochsenfeld effect describes the levitation of superconductors in a nonuniform magnetic field if they are cooled below critical temperature. This paper presents analytical solutions of the fractional London equation that describes the Meissner–Ochsenfeld effect considering the Liouville–Caputo, Caputo–Fabrizio–Caputo, Atangana–Baleanu–Caputo, fractional conformable derivative in Liouville–Caputo sense and Atangana–Koca–Caputo fractional-order derivatives. Numerical simulations were obtained for different values of the fractional-order.

scholarly journals Absorptive Repetitive Filters with Operators of Generalized Fractional Calculus

2013 ◽  
Vol 13 (4) ◽  
pp. 42-53 ◽  
Author(s):  
Nina Nikolova ◽  
Emil Nikolov
Keyword(s):  
Comparative Analysis ◽  
Fractional Calculus ◽  
Control Systems ◽  
Fractional Order ◽  
Fractional Integration ◽  
Frequency Characteristics ◽  
Rational Approximations ◽  
Integer Order ◽  
New Class ◽  
Generalized Fractional Calculus

Abstract : An essentially new class of repetitive fractional disturbance absorptive filters in disturbances absorbing control systems is proposed in the paper. Systematization of the standard repetitive fractional disturbance absorptive filters of this class is suggested. They use rational approximations of the operators for fractional integration in the theory of fractional calculus. The paper discusses the possibilities for repetitive absorbing of the disturbances with integer order filters and with fractional order filters. The results from the comparative analysis of their frequency characteristics are given below.

The Filtering Mechanism Modeling and Simulation of Passive Power Filter Based on Differentiator Circuit

2012 ◽  
Vol 430-432 ◽  
pp. 1593-1596
Author(s):  
Wan Neng Yu ◽  
Su Wen Li ◽  
Min Ying Zheng
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Integer Order ◽  
Filter Performance ◽  
Power Filter ◽  
Fractional Order Differential Equations ◽  
Passive Power ◽  
The Mathematical Model

Traditional continuous-time filters are of integer order which the power loss of passive power filter is general very much. However, using fractional calculus, filters may also be represented by the more general fractional-order differential equations. In this work, firstly, the passive elements were described with fractional-order differential equations depending on the introduction of fractional calculus application research. Secondly, the mathematical model of fractional-order filters was derived and discussed which includes high impedance at a certain frequency and low impedance at others, and the integer-order filters are only a tight subset of fractional-order filters that are testified. At last, the filter design idea to the fractional-order domain is developed and the better filter performance of the fractional-order passive power filter is validated by the mathematical model analysis and simulation results.

scholarly journals Can Fractional Calculus be Generalized: Problems and Efforts

2018 ◽  
Vol 11 (4) ◽  
pp. 1058-1099
Author(s):  
Syamal K. Sen ◽  
J. Vasundhara Devi ◽  
R.V.G. Ravi Kumar
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Fractional Order Calculus ◽  
Integer Order ◽  
Classical Calculus ◽  
Pros And Cons

Fractional order calculus always includes integer-order too. The question that crops up is: Can it be a widely accepted generalized version of classical calculus? We attempt to highlight the current problems that come in the way to define the fractional calculus that will be universally accepted as a perfect generalized version of integer-order calculus and to point out the efforts in this direction. Also, we discuss the question: Given a non-integer fractional order differential equation as a mathematical model can we readily write the corresponding physical model and vice versa in the same way as we traditionally do for classical differential equations? We demonstrate numerically computationally the pros and cons while addressing the questions keeping in the background the generalization of the inverse of a matrix.

scholarly journals 0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Baogui Xin ◽  
Yuting Li
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Three Dimensional ◽  
Financial System ◽  
Investment Incentive ◽  
Integer Order ◽  
System A ◽  
Weighted Integral ◽  
Test Algorithm ◽  
Predictor Corrector

A new integer-order chaotic financial system is extended by introducing a simple investment incentive into a three-dimensional chaotic financial system. A four-dimensional fractional-order chaotic financial system is presented by bringing fractional calculus into the new integer-order financial system. By using weighted integral thought, the fractional order derivative's economics meaning is given. The 0-1 test algorithm and the improved Adams-Bashforth-Moulton predictor-corrector scheme are employed to detect numerically the chaos in the proposed fractional order financial system.

Fractional-Order PID Controller of a Heating-Furnace System

2012 ◽  
Vol 490-495 ◽  
pp. 1145-1149 ◽  
Author(s):  
Yan Mei Wang ◽  
Yi Jie Liu ◽  
Rui Zhu ◽  
Yan Zhu Zhang
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Pid Control ◽  
Pid Controller ◽  
Control Method ◽  
Heating Furnace ◽  
Order Model ◽  
Integer Order ◽  
System A ◽  
Fractional Order Pid

This paper discusses the fractional-order controller of heating-furnace system, a new PID controller of heating-furnace system based on fractional calculus will be considered. Classical PID control method is also studied. Then, this paper presents the fractional-order PID control method based on integer-order model of heating-furnace system. Meanwhile, simulation study is done. Comparing the control methods and strategies of integer order model of the heating-furnace system, a conclusion is drawn that PID control based on fractional calculus is much more complex than that of integer order controller. Numerical simulations are used to illustrate the improvements of the proposed controller for the integer-order heating-furnace systems.

Chaos in Nonlinear Fractional Systems

2018 ◽  
pp. 333-403
Author(s):  
Nasr-eddine Hamri
Keyword(s):  
Fractional Order ◽  
Total Order ◽  
Fractional Order Systems ◽  
Fractional Systems ◽  
Integer Order ◽  
Fractional Models ◽  
The Subject ◽  
Fractional Differential ◽  
Hereditary Properties ◽  
Potential Use

The first steps of the theory of fractional calculus and some applications traced back to the first half of the nineteenth century, the subject only really came to life over the last few decades. A particular feature is that fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional models in comparison with classical integer-order models; another feature is that scientists have developed new models that involve fractional differential equations in mechanics, electrical engineering. Many scientists have become aware of the potential use of chaotic dynamics in engineering applications. With the development of the fractional-order algorithm, the dynamics of fractional order systems have received much attention. Chaos cannot occur in continuous integer order systems of total order less than three due to the Poincare-Bendixon theorem. It has been shown that many fractional-order dynamical systems behave chaotically with total order less than three.

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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