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Fractional Calculus and Applied Analysis

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Volume 16, Issue 1


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


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

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About the article

Published Online: 2012-12-27

Published in Print: 2013-03-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 16, Issue 1, Pages 123–141, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0009-8.

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