Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions

Liang, Xin; Gao, Feng; Yang, Xiao‐Jun; Xue, Yi

doi:10.3390/e20010078

Cited by 6 publications

(7 citation statements)

References 25 publications

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“…Please note that for δ = 0, a = 0 and β = 1 − α, Equation (8) retrieves the fractional derivative form of Caputo; see Equation 3. The Prabhakar fractional derivative has revealed a class of interesting behaviors in the context of the viscoelasticity theory [52] and in anomalous advection-dispersion transport [53]. On the other hand, the fractional Prabhakar derivative has been an efficient tool in physical models to approach the transition among anomalous diffusions [54,55].…”

Section: Preliminary Concepts About Tempered Fractional Calculusmentioning

confidence: 99%

Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting

Santos

2019

Physics

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In this work, we investigate a series of mathematical aspects for the fractional diffusion equation with stochastic resetting. The stochastic resetting process in Evans–Majumdar sense has several applications in science, with a particular emphasis on non-equilibrium physics and biological systems. We propose a version of the stochastic resetting theory for systems in which the reset point is in motion, so the walker does not return to the initial position as in the standard model, but returns to a point that moves in space. In addition, we investigate the proposed stochastic resetting model for diffusion with the fractional operator of Prabhakar. The derivative of Prabhakar consists of an integro-differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes a series of fractional operators such as Riemann–Liouville–Caputo. We present how the generalized model of stochastic resetting for fractional diffusion implies a rich class of anomalous diffusive processes, i.e., ⟨ ( Δ x ) 2 ⟩ ∝ t α , which includes sub-super-hyper-diffusive regimes. In the sequence, we generalize these ideas to the fractional Fokker–Planck equation for quadratic potential U ( x ) = a x 2 + b x + c . This work aims to present the generalized model of Evans–Majumdar’s theory for stochastic resetting under a new perspective of non-static restart points.

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Section: Preliminary Concepts About Tempered Fractional Calculusmentioning

confidence: 99%

Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting

Santos

2019

Physics

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“…We will also observe in passing that this paradox persists even for generalizations of entropy available in the literature such as Tsallis and Renyi entropies [4,6,[12][13][14]. In this context, we want to note that there might be implications of the entropy production paradox on applications known in finance [15,16], ecology [17], computational neuroscience [18], and physics [19][20][21].…”

Section: Introductionmentioning

confidence: 90%

Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

Hoffmann

Kulmus

Essex

et al. 2018

Entropy

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The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive increase of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.

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“…These mathematical tools are used to approximate the term d μ f (t)/dt μ (μ is a fractional order in a fractional differential problem) found in FOCP. The most commonly used approaches are presented in Podlubny (1999), Demirci and Ozalp (2012), Rehman and Khan (2012), Aslefallah and Rostamy (2014), Liu and Hou (2017), Yang et al (2017a), Yang et al (2017b), Li and Rui (2018), Liang et al (2018), Zhang (2018), and Yang et al (2019).…”

Section: Definitionsmentioning

confidence: 99%

“…The study of fractional calculus arises by the end of the seventeenth century. In the specialized literature, various contributions can be found, such as those on chemistry (Kirchner et al 2000), finance (Sabatelli et al 2002), hydrology (Schumer et al 2003), biology (Magin 2006), viscoelasticity (Larsson et al 2015), chaos synchronization (Su et al 2016), robotics (Kumar and Rana 2017), anomalous diffusion (Zhang et al 2017), anomalous heat-diffusion problems (Yang et al 2017a), advection-dispersion problems (Yuan et al 2016;Zhang, 2018;Li and Rui 2018), fractional Boussinesq equation (Yang et al 2017b), diffusion equations (Yang et al 2018), anomalous advection-dispersion equations (Liang et al 2018), solution of direct and inverse fractional advection-dispersion problem (Lobato et al 2019), and anomalous diffusion equations (Yang et al 2019).…”

Section: Introductionmentioning

confidence: 99%

Solution of Fractional Optimal Control Problems by using orthogonal collocation and Multi-objective Optimization Stochastic Fractal Search

2021

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In this contribution the solution of Fractional Optimal Control Problems (FOCP) by using the Orthogonal Collocation Method (OCM) and the Multi-objective Optimization Stochastic Fractal Search (MOSFS) algorithm is investigated. For this purpose, three classical case studies on engineering are considered. Initially, the concentration profiles of laccase enzyme production process are analyzed to evaluate the influence of fractional order. Then, two classical FOCP (Catalyst Mixing and Batch Reactor) are solved by using the association between OCM and MOSFS approachesthrough the formulation and solution of a multi-objective optimization problem. The results indicate that the variation of the fractional order impliesdifferent values for the original objective function. In addition, physicallyincoherent profiles can be obtained by considering the fluctuation of the fractional order. Finally, the proposed MOSFS is considered as apromising methodology to solve multi-objective optimization problems.

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Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions

Cited by 6 publications

References 25 publications

Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting

Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting

Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

Solution of Fractional Optimal Control Problems by using orthogonal collocation and Multi-objective Optimization Stochastic Fractal Search

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