Some aspects of fractional diffusion equations of single and distributed order
Abstract
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the firstorder time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is interpreted as a probability density of a selfsimilar nonMarkovian stochastic process related to a phenomenon of subdiffusion (the variance grows in time sublinearly). 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 nonMarkovian process that, however, is no longer selfsimilar but exhibits a corresponding distribution of timescales.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.4261
 Bibcode:
 2007arXiv0711.4261M
 Keywords:

 Mathematical Physics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 26A33;
 44A10;
 45K05;
 60G18;
 60J60
 EPrint:
 14 pages. International Symposium on "Analytic Function Theory, Fractional Calculus and Their Applications", University of Victoria (British Columbia, Canada), 2227 August 2005