scholarly journals Image Denoising Based on Adaptive Fractional Order with Improved PM Model

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Jimin Yu ◽  
Rumeng Zhai ◽  
Shangbo Zhou ◽  
LiJian Tan
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Diffusion Coefficients ◽  
Error Term ◽  
Noise Removal ◽  
Residual Error ◽  
Texture Information ◽  
Fractional Differential Operator ◽  
Proposed Model ◽  
Fractional Differential

In order to improve the image quality, in this paper, we propose an improved PM model. In the proposed model, we introduce two novel diffusion coefficients and a residual error term and replace the integer differential operator with the fractional differential operator in the PM model. The diffusion coefficients can be used effectively for edge detection and noise removal. The residual error term can help to prevent image distortion. Fractional order differential operator has a good characteristic that it can enhance image texture information while removing image noise. Additionally, in the two new diffusion coefficients, a novel method is proposed for automatically setting parameter k, and it does not need to do any experiments to get the value of k. For the computing fractional order diffusion coefficient, we employ the discrete Fourier transform, and an iterative scheme is carried out in the frequency domain. In the proposed model, not only is the integer differential operator replaced with the fractional differential operator, but also the order of the fractional differentiation is determined adaptively with the local variance. Comparing with some existing models, the experimental results show that the proposed algorithm can not only better suppress noise, but also better preserve edge and texture information. Moreover, the running time is greatly reduced.

scholarly journals On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. F. Imaga ◽  
S. A. Iyase
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Half Line ◽  
At Resonance ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Problem At Resonance

AbstractIn this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

scholarly journals Texture Enhancement Based on the Savitzky-Golay Fractional Differential Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hamid A. Jalab ◽  
Rabha W. Ibrahim
Keyword(s):  
Image Processing ◽  
Differential Operator ◽  
Fractional Order ◽  
Fractional Operators ◽  
Enhancement Technique ◽  
Fractional Differential Operator ◽  
Weight Window ◽  
Fractional Differential ◽  
Fractional Filter ◽  
The Given

Texture enhancement for digital images is the most important technique in image processing. The purpose of this paper is to design a texture enhancement technique using fractional order Savitzky-Golay differentiator, which leads to generalizing the Savitzky-Golay filter in the sense of the Srivastava-Owa fractional operators. By employing this generalized fractional filter, texture enhancement is introduced. Consequently, it calculates the generalized fractional order derivative of the given image using the sliding weight window over the image. Experimental results show that the operator can extract more subtle information and make the edges more prominent. In general, the capability of the generalized fractional differential will be high because it is sensitive to the subtle fluctuations of values of pixels.

An Improved Fractional Differential Algorithm and Noise Immunity Analysis

2011 ◽  
Vol 121-126 ◽  
pp. 710-714
Author(s):  
Yu Ran Liu ◽  
Ming Liang Hou
Keyword(s):  
Differential Operator ◽  
Noise Immunity ◽  
Differential Operators ◽  
Natural Images ◽  
Edge Information ◽  
Integer Order ◽  
Texture Information ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Differential Algorithm

Texture and edge information are equally important to identification of most of natural images. To the problem of the existing integer order differential operators can’t extract texture information form the images, this study developed a fractional differential algorithm, which can extract texture and marginal information simultaneously, based on settlement of the drift problem of fractional differential operator. Experimental results showed that our algorithm can not only extract texture information but also extract more edge information than the traditional algorithms. And to the image with Gauss noise, our algorithm also have noise immunity.

On the Influence of Fractional Derivative on Chaos Control of a New Fractional-Order Hyperchaotic System

2020 ◽  
pp. 115-132
Author(s):  
Ahmed Ezzat Mohamed Matouk
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Differential Operators ◽  
Equilibrium Points ◽  
Hyperchaotic System ◽  
Fractional Differential Operator ◽  
Quadratic Nonlinearities ◽  
Potential Applications ◽  
Non Local ◽  
Fractional Differential

The non-local fractional differential operators have potential applications in many fields of science and technology but especially in the field of dynamical systems. This chapter introduces a new hyperchaotic dynamical system involving non-local fractional differential operator with singular kernel (the Caputo type). The system involves three quadratic nonlinearities and also three equilibrium points. Existence of chaotic and hyperchaotic attractors has been illustrated. Based on Matouk's stability theory of four-dimensional fractional-order systems, the influence of the fractional differential operator on stabilizing the proposed system to its three steady states has been shown. Numerical results have been provided to verify the theoretical analysis. This kind of study is expected to add useful applications to chaos-based secure communications and text encryption.

scholarly journals A q-Weibull Counting Process through a Fractional Differential Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Kunnummal Muralidharan ◽  
Seema S. Nair
Keyword(s):  
Differential Operator ◽  
Weibull Distribution ◽  
Fractional Order ◽  
Counting Process ◽  
Fractional Differential Operator ◽  
Special Cases ◽  
Fractional Differential ◽  
Interarrival Times

We use the q-Weibull distribution and define a new counting process using the fractional order. As a consequence, we introduce a q-process with q-Weibull interarrival times. Some interesting special cases are also discussed which leads to a Mittag-Leffler form.

scholarly journals On Generalized Fractional Differentiator Signals

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Hamid A. Jalab ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Stirling Numbers ◽  
Signal Function ◽  
Closed Formula ◽  
Fractional Order Derivative ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Additive Combination ◽  
Fractional Differentiator

By employing the generalized fractional differential operator, we introduce a system of fractional order derivative for a uniformly sampled polynomial signal. The calculation of the bring in signal depends on the additive combination of the weighted bring-in ofNcascaded digital differentiators. The weights are imposed in a closed formula containing the Stirling numbers of the first kind. The approach taken in this work is to consider that signal function in terms of Newton series. The convergence of the system to a fractional time differentiator is discussed.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Rashida Zafar ◽  
Mujeeb ur Rehman ◽  
Moniba Shams
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Taylor Series ◽  
General Framework ◽  
Taylor Expansion ◽  
Fractional Integral ◽  
Haar Wavelet ◽  
Wavelet Method ◽  
Fractional Differential Operator ◽  
Fractional Differential

Abstract In this paper a general framework is presented on some operational properties of Caputo modification of Hadamard-type fractional differential operator along with a new algorithm proposed for approximation of Hadamard-type fractional integral using Haar wavelet method. Moreover, a generalized Taylor expansion based on Caputo–Hadamard-type fractional differential operator is also established, and an example is presented for illustration.

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