Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

Sene, Ndolane

doi:10.3390/fractalfract4020015

Cited by 10 publications

(5 citation statements)

References 33 publications

(94 reference statements)

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“…The Grünwald–Letnikov derivative is one of the proposed derivatives to answer Leibniz's question related to the definition of the fractional derivative (Sene, 2020), it can be regarded as the discretization of the fractional derivatives of Riemann‐Liouville and Caputo.…”

Section: Mathematical Modelingmentioning

confidence: 99%

Fractional derivative modeling for characterizing stress relaxation properties of Hami melon during drying

Yang

Zhou

et al. 2023

Journal of Texture Studies

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The viscoelastic properties of food materials will change significantly while drying is in progress, which greatly influences the food deformation caused by drying. This study aims to predict the viscoelastic mechanical behavior of Hami melon during drying using the fractional derivative model. To characterize the relaxation characteristics, based on the finite difference method, an improved Grünwald–Letnikov fractional stress relaxation model is proposed to derive an approximate discrete numerical solution of the relaxation modulus by applying the time fractional calculus. The Laplace transform method is used to verify the obtained results, and the equivalence of the two methods is proved. In addition, the stress relaxation tests prove that the fractional derivative model has a better prediction effect on the stress relaxation behavior of viscoelastic food than classical Zener model. The significant correlations between the fractional order and the stiffness coefficient and the moisture content is also studied. Which is negative correlation and positive correlation respectively.

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Section: Mathematical Modelingmentioning

confidence: 99%

Fractional derivative modeling for characterizing stress relaxation properties of Hami melon during drying

Yang

Zhou

et al. 2023

Journal of Texture Studies

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“…Fractional derivatives are widely used in fractal theory [5], diffusion theory [6], signal processing [7], and financial theory ( [8][9][10]) due to their non-locality advantages, i.e., the current state is influenced not only by the past instantaneous state but also by the state of the past period of time, which is more in line with the options market. Fractional derivative have two main forms: the Riemann-Liouville derivative [11] and the Caputo derivative [12].…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Calibration of European Option Volatility and Fractional Order under the Time Fractional Vasicek Model

Du,

2024

Algorithms

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In this paper, we recover the European option volatility function σ(t) of the underlying asset and the fractional order α of the time fractional derivatives under the time fractional Vasicek model. To address the ill-posed nature of the inverse problem, we employ Tikhonov regularization. The Alternating Direction Multiplier Method (ADMM) is utilized for the simultaneous recovery of the parameter α and the volatility function σ(t). In addition, the existence of a solution to the minimization problem has been demonstrated. Finally, the effectiveness of the proposed approach is verified through numerical simulation and empirical analysis.

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“…Currently, the development of research on fractions is so rapid, both grouped in Fractional Integrals and Derivatives [1,2], and in the Calculus Fractional group [3]. Fractional models are also very diverse, including: Diffusion-Reaction Equation model and Fractional Gas Dynamics Equation model [4,10]. The most widely used fractional operators are Riemann-Liouville and Caputo operators, only a few use Grundwald-Letnikov as in [5,6,13].…”

Section: Introductionmentioning

confidence: 99%

Grundwald-Letnikov Operator and Its Role in Solving Fractional Differential Equations

Parmikanti¹,

Rusyaman

2022

EKSAKTA

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Leibnitz in 1663 introduced the derivative notation for the order of natural numbers, and then the idea of fractional derivatives appeared. Only a century later, this idea began to be realized with the discovery of the concepts of fractional derivatives by several mathematicians, including Riemann (1832), Grundwal, Fourier, and Caputo in 1969. The concepts in the definitions of fractional derivatives by Riemann-Liouville and Caputo are more frequently used than other definitions, this paper will discuss the Grunwald-Letnikov (GL) operator, which has been discovered in 1867. This concept is less popular when compared to the Riemann-Liouville and Caputo concepts, however, this concept is quite interesting because the concept of derivation is developed from the definition of ordinary derivatives. In this paper will be shown that the formulas for the fractional derivative using the GL concept are the same as the results obtained using the Riemann-Liouville and Caputo concepts. As a complement, we will give an example of solving a fractional differential equation using Modified Homotopy Perturbation Methods.

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Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

Cited by 10 publications

References 33 publications

Fractional derivative modeling for characterizing stress relaxation properties of Hami melon during drying

Fractional derivative modeling for characterizing stress relaxation properties of Hami melon during drying

Simultaneous Calibration of European Option Volatility and Fractional Order under the Time Fractional Vasicek Model

Grundwald-Letnikov Operator and Its Role in Solving Fractional Differential Equations

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