Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4635833 | Applied Mathematics and Computation | 2007 | 11 Pages |
Abstract
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Francesco Mainardi, Gianni Pagnini, Rudolf Gorenflo,