Numerics of Fractional Langevin Equation Driven by Fractional Brownian Motion Using Non-Singular Fractional Derivative

2020 ◽  
Vol 6 (4) ◽  
pp. 253-262
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Fractional Brownian Motion ◽  
Langevin Equation ◽  
Fractional Langevin Equation

Numerics for the fractional Langevin equation driven by the fractional Brownian motion

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peng Guo ◽  
Caibin Zeng ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Order ◽  
Langevin Equation ◽  
Mean Square Displacement ◽  
Hurst Parameter ◽  
Mean Square ◽  
Fractional Langevin Equation ◽  
Normal Diffusion ◽  
The Mean

AbstractWe study analytically and numerically the fractional Langevin equation driven by the fractional Brownian motion. The fractional derivative is in Caputo’s sense and the fractional order in this paper is α = 2 − 2H, where H ∈ ($\tfrac{1} {2} $, 1) is the Hurst parameter (or, index). We give numerical schemes for the fractional Langevin equation with or without an external force. From the figures we can find that the mean square displacement of the fractional Langevin equation has the property of the anomalous diffusion. When the fractional order tends to an integer, the diffusion reduces to the normal diffusion.

A New Look at the Fractional Brownian Motion Definition

2007 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
Arnaldo Guimara˜es Batista
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives ◽  
Autocorrelation Functions ◽  
Fractional Noise ◽  
Simulation Procedure ◽  
Definition Of ◽  
Fractional Noises

A reinterpretation of the classic definition of fractional Brownian motion leads to a new definition involving a fractional noise obtained as a fractional derivative of white noise. To obtain this fractional noise, two sets of fractional derivatives are considered: a) the forward and backward and b) the central derivatives. For these derivatives the autocorrelation functions of the corresponding fractional noises have the same representations. The obtained results are used to define and propose a new simulation procedure.

scholarly journals Fractal Stochastic Processes on Thin Cantor-Like Sets

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 613
Author(s):  
Alireza Khalili Golmankhaneh ◽  
Renat Timergalievich Sibatov
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Fourier Transformation ◽  
One Dimensional ◽  
Fractal Derivative ◽  
Fractal Calculus ◽  
Non Local

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

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