Nonlinear Dynamics

, Volume 29, Issue 1–4, pp 129–143 | Cite as

Time Fractional Diffusion: A Discrete Random Walk Approach

  • Rudolf Gorenflo
  • Francesco Mainardi
  • Daniele Moretti
  • Paolo Paradisi


The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order β ∋ (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.

anomalous diffusion random walks fractional derivatives stochastic processes self-similarity 


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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Rudolf Gorenflo
    • 1
  • Francesco Mainardi
    • 2
  • Daniele Moretti
    • 3
  • Paolo Paradisi
    • 4
  1. 1.Erstes Mathematisches InstitutFreie Universität BerlinBerlinGermany
  2. 2.Dipartimento di FisicaUniversità di Bologna, and INFN, Sezione di BolognaBolognaItaly
  3. 3.CRIBISNET S.p.A.BolognaItaly
  4. 4.ISACIstituto di Scienze dell'Atmosfera e del Clima, Sezione di LecceLecceItaly

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