Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker–Planck equations

2004 ◽  
Vol 22 (4) ◽  
pp. 907-925 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Random Walk ◽  
Fractional Derivative ◽  
Complex Plane ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Shao-Hong Yan ◽  
Xiao-Hong Chen ◽  
Gong-Nan Xie ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Present Method ◽  
Decomposition Method ◽  
Cantor Set ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Fokker Planck Equations ◽  
Fokker Planck

The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

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

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Qianqian Yang ◽  
Fawang Liu ◽  
Ian Turner
Keyword(s):  
Fractional Derivative ◽  
Numerical Method ◽  
Source Term ◽  
Planck Equation ◽  
Stability And Convergence ◽  
Nonlinear Source ◽  
Transport Dynamics ◽  
Time Space ◽  
Fokker Planck Equations ◽  
Fokker Planck

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 (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order (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.

scholarly journals Fokker–Planck representations of non-Markov Langevin equations: application to delayed systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180131 ◽  
Author(s):  
Luca Giuggioli ◽  
Zohar Neu
Keyword(s):  
Langevin Equation ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Delay Systems ◽  
Langevin Equations ◽  
Delayed Systems ◽  
Temporal Features ◽  
Fokker Planck Equations ◽  
Random Fluctuations ◽  
Fokker Planck

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker–Planck equations for the n -time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n  = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker–Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

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