Random Walks Associated with Nonlinear Fokker–Planck Equations

Mendes, Renio dos Santos; Lenzi, E. K.; Malacarne, L. C.; Picoli, S.; Jauregui, Max

doi:10.3390/e19040155

Cited by 18 publications

(6 citation statements)

References 43 publications

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“…It is worth to note that the connections between stochastic processes and nonlinear Fokker-Planck equations have also been analyzed in [15,25] and in references therein.…”

Section: Introductionmentioning

confidence: 99%

Alternative probabilistic representations of Barenblatt-type solutions

Gregorio¹,

Garra²

2020

Modern Stochastics: Theory and Applications

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“…It is worth to note that the connections between stochastic processes and nonlinear Fokker-Planck equations have also been analyzed in [15,25] and in references therein.…”

Section: Introductionmentioning

confidence: 99%

Alternative probabilistic representations of Barenblatt-type solutions

Gregorio¹,

Garra²

2020

Modern Stochastics: Theory and Applications

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“…Complementary, nonlinear, heterogeneous and fractional approaches also provide generalisations of the FPE. Nonlinear diffusion equations, whose solutions exhibit a non-Gaussian behaviour, has been associated to nonlinear FPE in several scenarios: quantum walks [16], non-extensive statistical mechanics [17][18][19][20][21], generalisations of the Central Limit Theorem which imply non-Gaussian forms (called as q-Gaussian distributions) [22][23][24][25], among others. In addition, the study of the FPE with a nonconstant diffusion coefficient has allowed to introduce a natural generalisation of the Brownian motion in an heterogeneous medium [26,27].…”

Section: Introductionmentioning

confidence: 99%

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

Santos

Gomez

2018

J. Stat. Mech.

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Fractional diffusion equations imply non-Gaussian distributions that generalise the standard diffusive process. Recent advances in fractional calculus lead to a class of new fractional operators defined by non-singular memory kernels, differently from the fractional operator defined in the literature. In this work we propose a generalisation of the Fokker-Planck equation in terms of a non-singular fractional temporal operator and considering a non-constant diffusion coefficient. We obtain analytical solutions for the Caputo-Fabrizio and the Atangana-Baleanu fractional kernel operators, from which non-Gaussian distributions emerge having a long and short tails. In addition, we show that these non-Gaussian distributions are unimodal or bimodal according if the diffusion index ν is positive or negative respectively, where a diffusion coefficient of the power law type D(x) = D 0 |x| ν is considered. Thereby, a class of anomalous diffusion phenomena connected with fractional derivatives and with a diffusion coefficient of the power law type is presented. The techniques employed in this work open new possibilities for studying memory effects in diffusive contexts.

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“…A linear form of PME is the Fourier heat equation. The porous medium equation, also called by some authors NFPE [6], describes the diffusion of the molecules of a gas and fluid particles through porous media [7]. Despite the simplicity of PME, it is important to better understand this equation, because it is well known that most of the equations modeling physical phenomena without excessive simplification become nonlinear [8].…”

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confidence: 99%

Dual conformable derivative: Variational approach and nonlinear equations

Weberszpil

Godinho

Liang

2020

EPL

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In this letter, we present a variational approach with a recently proposed form of local deformed derivative, the dual conformable derivative, which leads us to obtain a class of nonlinear equations. The ansatz on the solutions can be mathematically inferred by this dual conformable derivative eigenvalue equation and the q-exponential family functions appear naturally. Also, a clear and natural justification for the appearance of the q → 2 - q-symmetry is given. To show the potential of our variational approach with this type of dual deformed derivative, we obtain the porous medium equation and present insight for the solution in terms of the q-Gaussian. Also, a new dual conformable wave equation, which is nonlinear, is proposed and a solution is built up in terms of q-plane waves. A dual conformable harmonic oscillator equation is also obtained and promptly solved by the natural ansatz. Aspects of the nonlinear Schroedinger equation are also contemplated and one shows that it can be obtained without the need of an additional ϕ-field, from a simple Lagrangian density. The solution to the nonlinear Schroedinger is also expressed in terms of the q-exponential family of functions.

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Random Walks Associated with Nonlinear Fokker–Planck Equations

Cited by 18 publications

References 43 publications

Alternative probabilistic representations of Barenblatt-type solutions

Alternative probabilistic representations of Barenblatt-type solutions

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

Dual conformable derivative: Variational approach and nonlinear equations

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