The time-fractional diffusion equation is obtained by generalizing the standard diffusion equation by using a proper time-fractional derivative of order \$1-beta\$ in the Riemann-Liouville (R-L) sense or of order \$beta\$ in the Caputo (C) sense, with \$beta in (0,1),.\$ The two forms are equivalent and the fundamental solution of the associated {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 two forms are no longer equivalent. However, the fundamental solution still is a probability density of a non-Markovian process but one exhibiting a distribution of time-scales instead of being self-similar: it is expressed in terms of an integral of Laplace type suitable for numerical computation. We consider with some detail two cases of diffusion of distributed order: the double order and the uniformly distributed order discussing the differences between the R-L and C approaches. For these cases we analyze in detail the behaviour of the fundamental solutions (numerically computed) and of the corresponding variance (analytically computed) through the exhibition of several plots. While for the R-L and for the C cases the fundamental solutions seem not to differ too much for moderate times, the behaviour of the corresponding variance for small and large times differs in a remarkable way.

### Sub-diffusion equations of fractional order and their fundamental solutions

#### Abstract

The time-fractional diffusion equation is obtained by generalizing the standard diffusion equation by using a proper time-fractional derivative of order \$1-beta\$ in the Riemann-Liouville (R-L) sense or of order \$beta\$ in the Caputo (C) sense, with \$beta in (0,1),.\$ The two forms are equivalent and the fundamental solution of the associated {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 two forms are no longer equivalent. However, the fundamental solution still is a probability density of a non-Markovian process but one exhibiting a distribution of time-scales instead of being self-similar: it is expressed in terms of an integral of Laplace type suitable for numerical computation. We consider with some detail two cases of diffusion of distributed order: the double order and the uniformly distributed order discussing the differences between the R-L and C approaches. For these cases we analyze in detail the behaviour of the fundamental solutions (numerically computed) and of the corresponding variance (analytically computed) through the exhibition of several plots. While for the R-L and for the C cases the fundamental solutions seem not to differ too much for moderate times, the behaviour of the corresponding variance for small and large times differs in a remarkable way.
##### Scheda breve Scheda completa Scheda completa (DC)
2007
Mathematical Methods in Engineering
23
55
F. Mainardi; A. Mura; G. Pagnini; R. Gorenflo
File in questo prodotto:
Eventuali allegati, non sono esposti

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/47136`
##### Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

• ND
• ND
• ND