A fractional Fokker–Planck equation for non-singular kernel operators

Santos, Maike A. F. dos; Gomez, Ignacio S.

doi:10.1088/1742-5468/aae5a2

Cited by 32 publications

(20 citation statements)

References 69 publications

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“…In fact, in this scenario, the Prabhakar derivative has been a mathematical tool that has been more understood day by day [38,58,59]. On the other hand, the Prabhakar derivative has as a particular case the Mittag-Leffler kernels with one and two parameters that have many applications in mathematical, physics, chemistry, and biology problems [35,36,[60][61][62].…”

Section: Preliminary Concepts About Tempered Fractional Calculusmentioning

confidence: 99%

“…Daily, fractional derivatives are applied in physics. Among other examples, there are chaotic systems [35], fractional FPE for non-singular kernels [29,36], the fractional Schrödinger equation [37], and viscoelasticity theory [38]. In addition, we want to emphasize that there is a class of derivatives that unifies the concept of tempered functions with the integral functions of Prabhakar.…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

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

Santos

2019

Physics

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“…Examples of these are: diffusion of charges in neurons [63], scattering patterns in the light spectrum [64], observation of anomalous diffusion and fractional self-similarity in one dimension [65], theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices [66], aging renewal theory and application to random walks [67]. In the references [68,69,60] we approaches issues related to fractional diffusion.…”

Section: The Walker Associated With Lévy Flightsmentioning

confidence: 99%

Analytic approaches of the anomalous diffusion: A review

Santos

2019

Chaos, Solitons & Fractals

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This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion, i.e. (∆x) 2 ∼ t α to α = 1. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.

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“…The convolution kernel in the diffusion equation is a typical manner of include memory effects [23,63,64]. The memory kernel can be put in model by two ways, the first one was discussed section 2, the second one was proposed in Ref.…”

Section: The Model With Memory Effectsmentioning

confidence: 99%

From continuous-time random walks to controlled-diffusion reaction

Santos¹

2019

J. Stat. Mech.

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Daily, are reported systems in nature that present anomalous diffusion phenomena due to irregularities of medium, traps or reactions process. In this scenario, the diffusion with traps or localised-reactions emerge through various investigations that include numerical, analytical and experimental techniques. In this work, we construct a model which involves a coupling of two diffusion equations to approach the random walkers in a medium with localised reaction point (or controlled diffusion). We present the exact analytical solutions to the model. In the following, we obtain the survival probability and mean square displacement. Moreover, we extend the model to include memory effects in reaction points. Thereby, we found a simple relation that connects the power-law memory kernels with anomalous diffusion phenomena, i.e. (x − x ) 2 ∝ t µ . The investigations presented in this work uses recent mathematical techniques to introduces a form to represent the coupled random walks in context of reaction-diffusion problem to localised reaction.

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A fractional Fokker–Planck equation for non-singular kernel operators

Cited by 32 publications

References 69 publications

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

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

Analytic approaches of the anomalous diffusion: A review

From continuous-time random walks to controlled-diffusion reaction

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