Abstract
We 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.P. Bouchaud, R. Cont, A Langevin approach to stock market fluctuations and crashes. Eur. Phys. J. B 6, No 4 (1998), 543–550.
A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag, New York (1997).
J.F. Coeurjolly, Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study. J. Stoch. Softw. 5, No 7 (2000), 1–53.
W.T. Coffey, Y.P. Kalmykov and J.T. Waldron, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, World Scientific Press, Singapore (2004).
C.H. Eab, S.C. Lim, Fractional generalized Langevin equation approach to single-file diffusion. Phys. A 389, No 13 (2010), 2510–2521.
K.S. Fa, Generalized Langevin equation with fractional derivative and long-time correlation function. Phys. Rev. E 73, No 6 (2006), 061104-1–061104-4.
K.S. Fa, Fractional Langevin equation and Riemann-Liouville fractional derivative. Eur. Phys. J. E 24, No 2 (2007), 139–143.
J.G.E.M. Fraaije, A.V. Zvelindovsky, G.J.A. Sevink and N.M. Maurits, Modulated self-organization in complex amphilic systems. Mol. Simul. 25, No 3–4 (2000), 131–144.
P. Guo, Numerical Simulations of the Fractional Differential Equations in Stochastics. Ph. D. disseration, Shanghai University (2012).
R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Press, Singapore (2000).
E.J. Hinch, Application of the Langevin equation to fluid suspensions. J. Fluid Mech. 72, No 3 (1975), 499–511.
F. Hu, W.Q. Zhu, L.C. Chen, Stochastic Hopf bifurcation of quasiintegrable Hamiltonian systems with fractional derivative damping. Int. J. Bifurcation and Chaos 22, No 4 (2012), 1250083-1–1250083-13.
A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Ltd., Netherlands (2006).
V. Kiryakova, Generalized Fractional Calculus and Applications. Longman Sci. & Technical and J. Wiley, Harlow — N. York (1994).
R.A. Kosinski, A. Grabowski, Langevin equations for modeling evacuation processes. Acta Phys. Pol. B 3, No 2 (2010), 365–377.
V. Kobelev, E. Romanov, Fractional Langevin equation to describe anomalous diffusion. Prog. Theor. Phys. 2000, No 139 (2000), 470–479.
R. Kubo, The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, No 1(1966), 255–284.
C.P. Li, F.H. Zeng, The finite difference methods for the fractional ordinary differential equations. Numer. Funct. Anal. Optimiz. 34, No 1 (2013), In press; DOI:10.1080/01630563.2012.706673.
C.P. Li, Z.G. Zhao, Introduction to fractional integrability and differentiability. Eur. Phys. J.-ST 193, No 1 (2011), 5–26.
C.P. Li, F.H. Zeng, F.W. Liu, Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 383–406; DOI:10.2478/s13540-012-0028-x; http://link.springer.com/article/10.2478/s13540-012-0028-x.
S.C. Lim, M. Li and L.P. Teo, Langevin equation with two fractional orders. Phys. Lett. A 372, No 42 (2008), 6309–6320.
E. Lutz, Fractional Langevin equation. Phys. Rev. E 64, No 5 (2001), 051106-1–051106-4.
F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, No 1–2 (2000), 283–299.
F. Mainardi, F. Tampieri, Diffusion regimes in Brownian motion induced by the Basset history force. Techn. Pap. No 1 (ISAO-TP-99/1), ISAO-CNR, Bologna, March 1999, pp. 25 (Inv. Lecture at Meeting of TAO, Working Group on Diffusion, Stockholm, Sweden, Oct. 1997).
B.B. Mandelbrot, J.W. Van Ness, Fractional Brownian motions, fractional noise and applications. SIAM. Rev. 10, No 4 (1968), 422–437.
K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience Publication, New York (1993).
K.B. Oldham, J. Spainer, The Fractional Calculus. Academic Press, New York (1974).
I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).
L.C.G. Rogers, Arbitrage with fractional Brownian motion. Math. Financ. 7, No 1 (1997), 95–105.
J. Schluttig, D. Alamanova, V. Helms and U.S. Schwarz, Dynamics of protein-protein encounter: A Langevin equation approach with reaction patches. J. Chem. Phys. 129, No 15 (2008), 155106-1–155106-1.
A. Takahashi, Low-Energy Nuclear Reactions and New Energy Technologies Sourcebook. Oxford University Press, Cary (2009).
B.J. West, S. Picozzi, Fractional Langevin model of memory in financial market. Phys. Rev. E 66, No 4 (2002), 037106-1–037106-12.
K. Wodkiewicz, M.S. Zubairy, Exact solution of a nonlinear Langevin equation with applications to photoelectron counting and noise-induced instability. J. Math. Phys. 24, No 6 (1983), 1401–1404.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editorial Note: For the presentation of this paper, the first author Peng Guo was awarded the prize for “Best Oral Presentation” at the 2012 Symposium on Fractional Differentiation and Its Applications (FDA’ 2012), Hohai University, Nanjing.
About this article
Cite this article
Guo, P., Zeng, C., Li, C. et al. Numerics for the fractional Langevin equation driven by the fractional Brownian motion. fcaa 16, 123–141 (2013). https://doi.org/10.2478/s13540-013-0009-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s13540-013-0009-8