Conformable derivative approach to anomalous diffusion

Zhou, Hongwei; Yang, Shuai; Zhang, S.Q.

doi:10.1016/j.physa.2017.09.101

Cited by 122 publications

(62 citation statements)

References 33 publications

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“…This derivative may not be seen as fractional derivative but has fractional compound. This new operator has attracted considerable attention in recent years [25][26][27][28][29][30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

Ghanbari

Gómez-Aguilar

2019

Rev. Mex. Fís.

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In this paper, the generalized exponential rational function method (GERFM) and the extended sinh-Gordon equation expansion method (ShGEEM) are used to construct exact solutions of the perturbed β-conformable-time Radhakrishnan-Kundu-Lakshmanan (RKL) equation. This model governs soliton propagation dynamics through a polarization-preserving fiber. Fractional derivatives are described in the β-conformable sense. As a result, we get new form of solitary traveling wave solutions for this model including novel soliton, traveling waves and kink-type solutions with complex structures. Physical interpretations of some extracted solutions are also included through taking suitable values of parameters and derivative order in them. It is proved that these methods are powerful, efficient, and can be fruitfully implemented to establish new solutions of nonlinear conformable-time partial differential equations applied in mathematical physics.

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Section: Introductionmentioning

confidence: 99%

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

Ghanbari

Gómez-Aguilar

2019

Rev. Mex. Fís.

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“…And some results have been obtained on the properties of the conformable fractional derivative [11][12][13]. Several applications and generalizations of the definition were also discussed in [14][15][16][17][18][19][20], among which [14] indicated that several specific conformable fractional models are consistent with experimental date, and which [15] interpreted the physical and geometrical meaning of the conformable fractional derivative. Although the definite meaning indicates potential applications of the conformable fractional derivative in physics and engineering, it is worth noting that the investigation of the theory of conformable fractional differential equations has only entered an initial stage.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of conformable fractional differential equations with integral boundary conditions

Wang

2018

Bound Value Probl

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In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation T α x(t) + f (t, x(t)) = 0, t ∈ [0, 1], subject to the boundary conditions x(0) = 0 and x(1) = λ 1 0 x(t) dt, where the order α belongs to (1, 2], T α x(t) denotes the conformable fractional derivative of a function x(t) of order α, and f : [0, 1] × [0, ∞) → [0, ∞) is continuous. By use of the fixed point theorem in a cone, some criteria for the existence of at least one positive solution are established. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem. A concrete example is given to illustrate the possible application of the obtained results.

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“…As stated in the literature, in some extent, these models above are suitable for the description of non-linear flow in porous media with lower permeability [9][10][11][12][13][14]. Most recently, Yang [14] and Zhou [15] also suggested that the conformable derivative approach is suitable for the describing non-linear flow in low-permeability porous media. However, these models above never took effective stress into account.…”

Section: Introductionmentioning

confidence: 95%

“…Until now, appropriate non-linear model for fluid flow through tight porous media remains unclear, though some more formulas have been established to describe the non-linear flow, such as power function model [9], exponential function model [10], incomplete Gamma function model [11,12] and fractional derivative approach [13][14][15]. As stated in the literature, in some extent, these models above are suitable for the description of non-linear flow in porous media with lower permeability [9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

A Non-Linear Flow Model for Porous Media Based on Conformable Derivative Approach

et al. 2018

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Prediction of the non-linear flow in porous media is still a major scientific and engineering challenge, despite major technological advances in both theoretical and computational thermodynamics in the past two decades. Specifically, essential controls on non-linear flow in porous media are not yet definitive. The principal aim of this paper is to develop a meaningful and reasonable quantitative model that manifests the most important fundamental controls on low velocity non-linear flow. By coupling a new derivative with fractional order, referred to conformable derivative, Swartzendruber equation and modified Hertzian contact theory as well as fractal geometry theory, a flow velocity model for porous media is proposed to improve the modeling of Non-linear flow in porous media. Predictions using the proposed model agree well with available experimental data. Salient results presented here include (1) the flow velocity decreases as effective stress increases; (2) rock types of “softer” mechanical properties may exhibit lower flow velocity; (3) flow velocity increases with the rougher pore surfaces and rock elastic modulus. In general, the proposed model illustrates mechanisms that affect non-linear flow behavior in porous media.

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Conformable derivative approach to anomalous diffusion

Cited by 122 publications

References 33 publications

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

Positive solutions of conformable fractional differential equations with integral boundary conditions

A Non-Linear Flow Model for Porous Media Based on Conformable Derivative Approach

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