Fractional Fokker-Planck Equation with Space and Time Dependent Drift and Diffusion

Lv, Longjin; Qiu, Weiyuan; Ren, Fu-Yao

doi:10.1007/s10955-012-0618-3

Cited by 14 publications

(24 citation statements)

References 20 publications

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“…This problem was also studied in [2][3][4][5][6], where the author investigated solution of the anomalous diffusion equation with space-time dependent forces. In this paper we show that the result of [1] is true in even more general setting. Namely, the factorization of the space-time dependent drift and diffusion coefficients into a space dependent part and a time dependent part is not necessary.…”

Section: Introductionmentioning

confidence: 60%

“…Namely, the factorization of the space-time dependent drift and diffusion coefficients into a space dependent part and a time dependent part is not necessary. We also point out some drawbacks in the derivation of the main result in [1] and present the correct proof.…”

Section: Introductionmentioning

confidence: 77%

“…In recent paper by Lv et al [1] authors studied the question of stochastic representation of anomalous diffusion with space and time dependent drift and diffusion coefficient. This problem was also studied in [2][3][4][5][6], where the author investigated solution of the anomalous diffusion equation with space-time dependent forces.…”

Section: Introductionmentioning

confidence: 99%

“…In [1] authors analyzed the following fractional Fokker-Planck equation (FFPE) Moreover, the operator in braces is the classical Fokker-Planck operator with F(x, t) and D(x, t) being the drift and diffusion parameters, respectively. Equation (1) describes the evolution in time of the the probability density function (PDF) p(x, t) of some subdiffusive process with sublinear in time mean square displacement.…”

Section: Introductionmentioning

confidence: 99%

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Comment on Fractional Fokker–Planck Equation with Space and Time Dependent Drift and Diffusion

2014

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The purpose of this paper is to correct errors presented recently in the paper [Lv et al. J Stat Phys 149:619-628 (2012)], where the authors analyzed Fractional Fokker- Planck equation (FFPE) with space-time dependent drift F(x, t) = F(x) f (t) and diffusion D(x, t) = D(x)d(t)coefficients in the factorized form. We show an important drawback in the derivation of the stochastic representation of FFPE presented in the aforementioned paper, which makes the whole proof wrong. Moreover, we present a correct proof of their result in even more general case, when both drift and diffusion can have any, not necessarily factorized, form.

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Comment on Fractional Fokker–Planck Equation with Space and Time Dependent Drift and Diffusion

2014

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“…In recent works the time fractional Fokker-Planck equation with space and time dependent force and diffusion has been studied, such as, in [7,13,14], where physical and stochastic interpretations have been analyzed. In [7,24] this type of equation has been discussed using Langevin and continuous-time random walk approaches, which clarified some of the issues addressed in [6] for time dependent coefficients.…”

Section: The Modelmentioning

confidence: 99%

Numerical solution of a time-space fractional Fokker Planck equation with variable force field and diffusion

Pinto

Sousa

2017

Communications in Nonlinear Science and Numerical Simulation

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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

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Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

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Fractional Fokker-Planck Equation with Space and Time Dependent Drift and Diffusion

Cited by 14 publications

References 20 publications

Comment on Fractional Fokker–Planck Equation with Space and Time Dependent Drift and Diffusion

Comment on Fractional Fokker–Planck Equation with Space and Time Dependent Drift and Diffusion

Numerical solution of a time-space fractional Fokker Planck equation with variable force field and diffusion

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

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