Complex Fractional Moments for the Characterization of the Probabilistic Response of Non-linear Systems Subjected to White Noises

  • Mario Di Paola
  • Antonina PirrottaEmail author
  • Gioacchino Alotta
  • Alberto Di Matteo
  • Francesco Paolo Pinnola
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 228)


In this chapter the solution of Fokker-Planck-Kolmogorov type equations is pursued with the aid of Complex Fractional Moments (CFMs). These quantities are the generalization of the well-known integer-order moments and are obtained as Mellin transform of the Probability Density Function (PDF). From this point of view, the PDF can be seen as inverse Mellin transform of the CFMs, and it can be obtained through a limited number of CFMs. These CFMs’ capability allows to solve the Fokker-Planck-Kolmogorov equation governing the evolutionary PDF of non-linear systems forced by white noise with an elegant and efficient strategy. The main difference between this new approach and the other one based on integer moments lies in the fact that CFMs do not require the closure scheme because a limited number of them is sufficient to accurately describe the evolutionary PDF and no hierarchy problem occurs.


Probability density function Fokker-Planck equation Complex fractional moments 


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Mario Di Paola
    • 1
  • Antonina Pirrotta
    • 1
    Email author
  • Gioacchino Alotta
    • 1
  • Alberto Di Matteo
    • 1
  • Francesco Paolo Pinnola
    • 2
  1. 1.Department of EngineeringUniversity of PalermoPalermoItaly
  2. 2.Department of Structures for Engineering and Architecture (DIST)University of Naples “Federico II”NaplesItaly

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