An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type

Liang, Xin; Gao, Feng; Zhou, Chunbo; Wang, Zhen; Yang, Xiao‐Jun

doi:10.1186/s13662-018-1478-1

Cited by 18 publications

(3 citation statements)

References 38 publications

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“…where α is the fractional order which can be a complex number, R(α) denotes the real part of α and a < t, a is the fixed lower terminal and t is the moving upper terminal. There exist several definitions for fractional derivatives and fractional integrals like the Riemann-Liouville, Caputo, Hadamard, Riesz, Griinwald-Letnikov [52][53][54][55]. The two most commonly used are Riemann-Liouville and Caputo definitions.…”

Section: Properties Of Fractional Derivativementioning

confidence: 99%

Synchronization of different order fractional-order chaotic systems using modify adaptive sliding mode control

Al-Sawalha

2020

Adv Differ Equ

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This paper proposes a modified adaptive sliding-mode control technique and investigates the reduced-order and increased-order synchronization between two different fractional-order chaotic systems using the master and slave system synchronization arrangement. The parameters of the master and slave systems are different and uncertain. These systems exhibit different chaotic behavior and topological properties. The dynamic behavior of the proposed synchronization schemes is more complex and unpredictable. These attributes of the proposed synchronization schemes enhance the security of the information signal in digital communication systems. The proposed switching law ensures the convergence of the error vectors to the switching surface and the feedback control signals guarantee the fast convergence of the error vectors to the origin. Lyapunov stability theory proves the asymptotic stability of the closed-loop. The paper also designs suitable parameters update laws the estimate the unknown parameters. Computer-based simulation results verify the theoretical findings.

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Section: Properties Of Fractional Derivativementioning

confidence: 99%

Synchronization of different order fractional-order chaotic systems using modify adaptive sliding mode control

Al-Sawalha

2020

Adv Differ Equ

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“…However, there are many phenomena that may not depend only on the time moment but also on the former time history, which cannot be modeled utilizing the classical derivatives. For this reason, many authors try to replace the classical derivatives with the so-called fractional derivatives in numerous contributions [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], because it has been proven that this last kind of derivatives is a very good way to describe processes with memory. According to the literature of fractional calculus, it is remarkable that there are many approaches to defining fractional derivatives, and each definition has advantages compared to others [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

Atraoui

Bouaouid

2021

Adv Differ Equ

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In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$ d α d t α [ d α x ( t ) d t α ] = A x ( t ) + f ( t , x ( t ) ) , $t\in [0,\tau ]$ t ∈ [ 0 , τ ] subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ x ( 0 ) = x 0 + g ( x ) and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ d α x ( 0 ) d t α = x 1 + h ( x ) , where $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ d α ( ⋅ ) d t α is the conformable fractional derivative of order $\alpha \in\, ]0,1]$ α ∈ ] 0 , 1 ] and A is the infinitesimal generator of a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ ( { C ( t ) , S ( t ) } ) t ∈ R on a Banach space X. The elements $x_{0}$ x 0 and $x_{1}$ x 1 are two fixed vectors in X, and f, g, h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ ( S ( t ) ) t > 0 and any Lipschitz conditions on the functions g and h.

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“…There are many definitions of fractional derivatives (three popular definitions were given by Grunwald-Letnikov (G-L), Riemann-Liouville (R-L), and Caputo). These have been used in numerous fields of science such as study of the anomalous diffusion phenomenon [24][25][26], circuit theory [27][28][29], and image processing [30,31], among other applications [11,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. Given the discussion above, we consider that using anisotropic diffusion models to eliminate noise in an image, preserving both strong and weak edges and without phenomena such as staircase, speckle, or any type of artifact, is a subject where much remains to be investigated.…”

Section: Introduction and Some Basic Definitionsmentioning

confidence: 99%

Perona-Malik Model with Diffusion Coefficient Depending on Fractional Gradient via Caputo-Fabrizio Derivative

Nchama¹,

Mecías²,

Ricard³

2020

Abstract and Applied Analysis

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The Perona-Malik (PM) model is used successfully in image processing to eliminate noise while preserving edges; however, this model has a major drawback: it tends to make the image look blocky. This work proposes to modify the PM model by introducing the Caputo-Fabrizio fractional gradient inside the diffusivity function. Experiments with natural images show that our model can suppress efficiently the blocky effect. Also, our model has good performance in visual quality, high peak signal-to-noise ratio (PSNR), and lower value of mean absolute error (MAE) and mean square error (MSE).

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An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type

Cited by 18 publications

References 38 publications

Synchronization of different order fractional-order chaotic systems using modify adaptive sliding mode control

Synchronization of different order fractional-order chaotic systems using modify adaptive sliding mode control

On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

Perona-Malik Model with Diffusion Coefficient Depending on Fractional Gradient via Caputo-Fabrizio Derivative

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