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Fractional Calculus and Applied Analysis

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

Peng Guo
  • Dept. of Mathematics, Shanghai University, Shangda Rd. 99, Shanghai, 200444, PR China
  • Dept. of Mathematics and Physics, Shanghai DianJi University, Ganlan Rd. 1350, Shanghai, 201306, PR China
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/ Caibin Zeng / Changpin Li / YangQuan Chen
Published Online: 2012-12-27 | DOI: https://doi.org/10.2478/s13540-013-0009-8

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.

MSC: 26A33 (main) 65L12; 60G22; 34A08; 39A70

Keywords: fractional Langevin equation; numerical algorithm; mean square displacement; fluctuation dissipation; Caputo derivative

  • [1] J.P. Bouchaud, R. Cont, A Langevin approach to stock market fluctuations and crashes. Eur. Phys. J. B 6, No 4 (1998), 543–550. http://dx.doi.org/10.1007/s100510050582CrossrefGoogle Scholar

  • [2] A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag, New York (1997). Google Scholar

  • [3] 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. Google Scholar

  • [4] 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). Google Scholar

  • [5] C.H. Eab, S.C. Lim, Fractional generalized Langevin equation approach to single-file diffusion. Phys. A 389, No 13 (2010), 2510–2521. http://dx.doi.org/10.1016/j.physa.2010.02.041CrossrefGoogle Scholar

  • [6] 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. http://dx.doi.org/10.1103/PhysRevE.73.061104Google Scholar

  • [7] K.S. Fa, Fractional Langevin equation and Riemann-Liouville fractional derivative. Eur. Phys. J. E 24, No 2 (2007), 139–143. http://dx.doi.org/10.1140/epje/i2007-10224-2CrossrefGoogle Scholar

  • [8] 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. http://dx.doi.org/10.1080/08927020008044119CrossrefGoogle Scholar

  • [9] P. Guo, Numerical Simulations of the Fractional Differential Equations in Stochastics. Ph. D. disseration, Shanghai University (2012). Google Scholar

  • [10] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Press, Singapore (2000). http://dx.doi.org/10.1142/9789812817747CrossrefGoogle Scholar

  • [11] E.J. Hinch, Application of the Langevin equation to fluid suspensions. J. Fluid Mech. 72, No 3 (1975), 499–511. http://dx.doi.org/10.1017/S0022112075003102CrossrefGoogle Scholar

  • [12] 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. Web of ScienceGoogle Scholar

  • [13] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Ltd., Netherlands (2006). Google Scholar

  • [14] V. Kiryakova, Generalized Fractional Calculus and Applications. Longman Sci. & Technical and J. Wiley, Harlow — N. York (1994). Google Scholar

  • [15] R.A. Kosinski, A. Grabowski, Langevin equations for modeling evacuation processes. Acta Phys. Pol. B 3, No 2 (2010), 365–377. Google Scholar

  • [16] V. Kobelev, E. Romanov, Fractional Langevin equation to describe anomalous diffusion. Prog. Theor. Phys. 2000, No 139 (2000), 470–479. Google Scholar

  • [17] R. Kubo, The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, No 1(1966), 255–284. http://dx.doi.org/10.1088/0034-4885/29/1/306CrossrefGoogle Scholar

  • [18] 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. CrossrefGoogle Scholar

  • [19] C.P. Li, Z.G. Zhao, Introduction to fractional integrability and differentiability. Eur. Phys. J.-ST 193, No 1 (2011), 5–26. http://dx.doi.org/10.1140/epjst/e2011-01378-2CrossrefGoogle Scholar

  • [20] 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. CrossrefGoogle Scholar

  • [21] S.C. Lim, M. Li and L.P. Teo, Langevin equation with two fractional orders. Phys. Lett. A 372, No 42 (2008), 6309–6320. http://dx.doi.org/10.1016/j.physleta.2008.08.045CrossrefGoogle Scholar

  • [22] E. Lutz, Fractional Langevin equation. Phys. Rev. E 64, No 5 (2001), 051106-1–051106-4. http://dx.doi.org/10.1103/PhysRevE.64.051106Google Scholar

  • [23] F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, No 1–2 (2000), 283–299. http://dx.doi.org/10.1016/S0377-0427(00)00294-6CrossrefGoogle Scholar

  • [24] 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). Google Scholar

  • [25] B.B. Mandelbrot, J.W. Van Ness, Fractional Brownian motions, fractional noise and applications. SIAM. Rev. 10, No 4 (1968), 422–437. http://dx.doi.org/10.1137/1010093CrossrefGoogle Scholar

  • [26] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience Publication, New York (1993). Google Scholar

  • [27] K.B. Oldham, J. Spainer, The Fractional Calculus. Academic Press, New York (1974). Google Scholar

  • [28] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999). Google Scholar

  • [29] L.C.G. Rogers, Arbitrage with fractional Brownian motion. Math. Financ. 7, No 1 (1997), 95–105. http://dx.doi.org/10.1111/1467-9965.00025CrossrefGoogle Scholar

  • [30] 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. http://dx.doi.org/10.1063/1.2996082CrossrefGoogle Scholar

  • [31] A. Takahashi, Low-Energy Nuclear Reactions and New Energy Technologies Sourcebook. Oxford University Press, Cary (2009). Google Scholar

  • [32] B.J. West, S. Picozzi, Fractional Langevin model of memory in financial market. Phys. Rev. E 66, No 4 (2002), 037106-1–037106-12. Google Scholar

  • [33] 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. http://dx.doi.org/10.1063/1.525874CrossrefGoogle Scholar

About the article

Published Online: 2012-12-27

Published in Print: 2013-03-01


Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0009-8.

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