The Grünwald–Letnikov Fractional-Order Derivative with Fixed Memory Length

Abdelouahab, Mohammed-Salah; Hamri, Nasr-Eddine

doi:10.1007/s00009-015-0525-3

Cited by 43 publications

(19 citation statements)

References 23 publications

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“…The definition of the G-L fractional derivative is derived from the derivative of the nthorder continuous function, (8,9) which is obtained by extending the order n to any real order a. Let the function f(x) be a continuous function, and there is a continuous derivative of order n, so the integer derivative of the 1st and 2nd orders can be defined as…”

Section: Definition Of Fractional Derivative and Smlrmentioning

confidence: 99%

Estimation of Available Phosphorus in Man Interference Zone Based on Fractional Derivative

An-Hong¹,

Cheng-Biao²,

Xiong³

2020

Sensors and Materials

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The fractional derivative has the advantages in terms of memory and globality, and it can overcome the shortcomings of the traditional integer differential algorithm. Moreover, the absorption characteristics of available phosphorus in soil in visible near-infrared bands are unclear, and the prediction model has a low precision. In this paper, we propose a novel method to improve the accuracy of the prediction model for available phosphorus content, which is based on the fractional derivative and stepwise multiple linear regression (SMLR). First, the relationship between the soil spectrum and the available phosphorus content under different fractional orders was studied. Secondly, spectrum dimensionality reduction based on sensitive bands was performed. Finally, the SMLR model was adopted to quantitatively predict the available phosphorus content, and the precision of different fractional order models was discussed. Simulation results revealed that the fractional derivative can describe the small differences in spectral data and increase the correlation between the soil spectrum and the available phosphorus content. The 1.4th-order model is the optimum fractional model. Thus, these results indicate that the fractional derivative could improve the accuracy of the estimation model for available phosphorus content.

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Section: Definition Of Fractional Derivative and Smlrmentioning

confidence: 99%

Estimation of Available Phosphorus in Man Interference Zone Based on Fractional Derivative

An-Hong¹,

Cheng-Biao²,

Xiong³

2020

Sensors and Materials

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“…1. The dynamic behaviors of interacting prey (x) and predator (y) for the order α = 0.7 with the initial values (x 0 , y 0 ) = (50,40), (50,55), (75,55), (80,45) are shown in Fig. 1(a).…”

Section: The Effect Of Different Parameter Valuesmentioning

confidence: 99%

“…1(a). For comparison, the relationships of the two species for α = 0.9 with (x 0 , y 0 ) = (51,45), (53,48), (75,50), (70,47) are shown in Fig. 1(b).…”

Section: The Effect Of Different Parameter Valuesmentioning

confidence: 99%

“…The most famous definitions of fractional order derivatives are the Grünwald-Letnikov [75], Riemann-Liouville [76], and Caputo [77] definitions. The Caputo definition is very useful because in this case the derivative of a constant is zero and the initial conditions for the fractional order differential equations can be provided in the same manner as for the classical integer case, which has a clear physical meaning [78].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

Baisad

Moonchai

2018

Adv Differ Equ

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The Kolmogorov model has been applied to many biological and environmental problems. We are particularly interested in one of its variants, that is, a Gauss-type predator-prey model that includes the Allee effect and Holling type-III functional response. Instead of using classic first order differential equations to formulate the model, fractional order differential equations are utilized. The existence and uniqueness of a nonnegative solution as well as the conditions for the existence of equilibrium points are provided. We then investigate the local stability of the three types of equilibrium points by using the linearization method. The conditions for the existence of a Hopf bifurcation at the positive equilibrium are also presented. To further affirm the theoretical results, numerical simulations for the coexistence equilibrium point are carried out.

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“…In [6], it is proposed a solution for this problem by imposing a simple modification to the Grünwald-Letnikov definition.…”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation and Chaos in Simplest Fractional-Order Memristor-based Electrical Circuit

Abdelouahab¹,

Lozi²

2015

Indian Jour. of Indust. and Appl. Math.

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Abstract:In this article, we investigate the bifurcation and chaos in a simplest fractional-order memristor-based electrical circuit composed of only three circuit elements: a linear passive capacitor, a linear passive inductor and a non-linear active memristor with two-degree polynomial memristance and a second-order exponent internal state. It is shown that this fractional circuit can exhibit a drastically rich non-linear dynamics such as a Hopf bifurcation, coexistence of two, three and four limit cycles, double-scroll chaotic attractor, four-scroll chaotic attractor, coexistence of one (or two) chaotic attractor with one limit cycle and new chaotic attractor which is not observed in the integer case. Finally, the presence of chaos is confirmed by the application of the recently introduced 0-1 test.

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The Grünwald–Letnikov Fractional-Order Derivative with Fixed Memory Length

Cited by 43 publications

References 23 publications

Estimation of Available Phosphorus in Man Interference Zone Based on Fractional Derivative

Estimation of Available Phosphorus in Man Interference Zone Based on Fractional Derivative

Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

Hopf Bifurcation and Chaos in Simplest Fractional-Order Memristor-based Electrical Circuit

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