2006/10/12 by Francesco Mainardi, Gianni Pagnini, Rudolf Gorenflo · 2 citations
Mathematics · Physics and Astronomy · #Differential Equations and Numerical Methods #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #cond-mat.dis-nn #cond-mat.stat-mech #math-ph #math.MP #msc:26A33 #msc:44A10 #msc:45K05 #msc:60G18 #msc:60J60
paper · pdf · doi:10.1016/j.amc.2006.08.126
published as Applied Mathematics and Computation, Vol. 187, No 1, pp. 295-305 (2007) · 14 pages. International Symposium on "Analytic Function Theory, Fractional Calculus and Their Applications", University of Victoria (British Columbia, Canada), 22-27 August 2005
openalex publication_date 2006/10/12 · arxiv created 2007/11/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order β∈ (0,1). The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of sub-diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales.