Fokker–Planck equation and subdiffusive fractional Fokker–Planck equation of bistable systems with sinks

Chow, Chit; Liu, K.L

doi:10.1016/j.physa.2004.04.114

Cited by 8 publications

(3 citation statements)

References 39 publications

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“…Fractional Fokker-Planck equation provides a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion [2] and nonexponential relaxation patterns [11]. It has been used to model dynamics of protein systems and for reactions occurring in disordered media [2,[12][13][14][15][16][17][18]. Description equivalent to a fractional Fokker-Planck equation consist of a Markovian dynamics governed by an ordinary Langevin equation but proceeding in an auxiliary, operational time instead of the physical time [19].…”

Section: Introductionmentioning

confidence: 99%

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Kazakevičius

Ruseckas

2015

Physica A: Statistical Mechanics and its Applications

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Subdiffusive behavior of one-dimensional stochastic systems can be described by timesubordinated Langevin equations. The corresponding probability density satisfies the time-fractional Fokker-Planck equations. In the homogeneous systems the power spectral density of the signals generated by such Langevin equations has power-law dependency on the frequency with the exponent smaller than 1. In this paper we consider nonhomogeneous systems and show that in such systems the power spectral density can have power-law behavior with the exponent equal to or larger than 1 in a wide range of intermediate frequencies.

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Section: Introductionmentioning

confidence: 99%

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Kazakevičius

Ruseckas

2015

Physica A: Statistical Mechanics and its Applications

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“…When the fractional differential equation is used to describe the asymptotic behavior of continuous time random walks, its solution corresponds to the Lévy walks, generalizing the Brownian motion to the Lévy motion. The following space fractional Fokker-Planck equation has been considered 2, 3, 7 : As a model for subdiffusion in the presence of an external field, a time fractional extension of the FPE has been introduced as the time fractional Fokker-Planck equation TFFPE 5,9 :…”

Section: Introductionmentioning

confidence: 99%

Stability and Convergence of an Effective Numerical Method for the Time‐Space Fractional Fokker‐Planck Equation with a Nonlinear Source Term

Yang

Liu

Turner

2010

International Journal of Differential Equations

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Fractional Fokker-Planck equations FFPEs have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term TSFFPENST , which involve the Caputo time fractional derivative CTFD of order α ∈ 0, 1 and the symmetric Riesz space fractional derivative RSFD of order μ ∈ 1, 2 . Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

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“…Being an extension of the classical Fokker-Planck equation, the SFFPE can be derived by several approaches [4][5][6][7], and it can also be regarded as the overdamped case of the fractional Klein-Kramers equation [2,3,7,8]. Recently, this equation has been applied to a number of problems which reveal the significance of subdiffusive dynamics, including dynamics of annealed systems [9], linear response in complex systems [10], diffusion on comb-like structure [11], subdiffusive motion of bistable systems [12], anomalous transport in tilted periodic potentials [13], subdiffusive reaction-diffusion processes [14][15][16][17][18][19][20] and fluorescence lifetime for small single molecules [21,22].…”

Section: Introductionmentioning

confidence: 99%

The response of subdiffusive bistable fractional Fokker–Planck systems to rectangular signals

Yim¹,

Liu

2007

J. Phys. A: Math. Theor.

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We study the response of one-dimensional subdiffusive fractional Fokker–Planck systems with a general confining potential, when it is perturbed from its stationary state by a time-dependent non-sinusoidal driving force. Three types of rectangular driving signals have been investigated: a rectangular pulse, a periodic telegraph signal and a generalized telegraph signal with a fractional duty cycle. We derive analytic expressions for the linear response and the input energy in one period of the driving signal. In particular, for signals with a long period, we obtain several asymptotic results concerning the wave form of the response and the stochastic energetics. Numerical results for representative symmetric, as well as asymmetric, subdiffusive bistable systems are presented and discussed.

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Fokker–Planck equation and subdiffusive fractional Fokker–Planck equation of bistable systems with sinks

Cited by 8 publications

References 39 publications

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Stability and Convergence of an Effective Numerical Method for the Time‐Space Fractional Fokker‐Planck Equation with a Nonlinear Source Term

The response of subdiffusive bistable fractional Fokker–Planck systems to rectangular signals

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