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Statistical analysis for stochastic systems including fractional derivatives

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Abstract

An analytical scheme to determine the statistical behavior of a stochastic system including two terms of fractional derivative with real, arbitrary, fractional orders is proposed. In this approach, Green’s functions obtained are based on a Laplace transform approach and the weighted generalized Mittag–Leffler function. The responses of the system can be subsequently described as a Duhamel integral-type close-form expression. These expressions are applied to obtain the statistical behavior of a dynamical system excited by stationary stochastic processes. The numerical simulation based on the modified Euler method and Monte Carlo approach is developed. Three examples of single-degree-of-freedom system with fractional derivative damping under Gaussian white noise excitation are presented to illustrate application of the proposed method.

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Authors and Affiliations

  1. Department of Mechanics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Z. L. Huang, X. L. Jin & Y. Wang

  2. Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong, People’s Republic of China

    C. W. Lim

Authors
  1. Z. L. Huang
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  2. X. L. Jin
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  3. C. W. Lim
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  4. Y. Wang
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Correspondence to C. W. Lim.

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Huang, Z.L., Jin, X.L., Lim, C.W. et al. Statistical analysis for stochastic systems including fractional derivatives. Nonlinear Dyn 59, 339–349 (2010). https://doi.org/10.1007/s11071-009-9543-7

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