Geodesic
3-dimensional physically consistent diffusion in anisotropic media with memory
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 2, pp. 131-143.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Some data on the flow of fluids exhibit properties which may not be interpreted with the classic theory of propagation of pressure and of fluids [21] based on the classic D’Arcy’s law which states that the flux is proportional to the pressure gradient. In order to obtain a better representation of the flow and of the pressure of fluids the law of D’Arcy is here modified introducing a memory formalisms operating on the flow as well as on the pressure gradient which implies a filtering of the pressure gradient without singularities; the properties of the filtering are also described. We shall also modify the second constitutive equation of diffusion, which relates the density variations of the fluid to its pressure variations, by introducing the rheology of the fluid also represented by derivatives of fractional order operating on the pressure as well as on the density. Moreover the medium will be considered anisotropic. We shall obtain the diffusion equation with these conditions in an anisotropic medium and find the Green function for a point source.
Alcuni dati sperimentali sul flusso di fluidi in mezzi porosi mostrano proprietà che non possono essere spiegate in base alla legge di D’Arcy che stabilisce proporzionalità fra flusso e gradiente di pressione. Per spiegare questa fenomenologia in questa Nota si modifica la legge di D’Arcy introducendo un formalismo di memoria, che opera sia sul flusso che sul gradiente di pressione, che genera un filtraggio senza singolarità fisicamente inaccettabili. Nella Nota si modifica anche l’equazione che lega la variazione di pressione con quella del fluido mediante l’introduzione di un secondo formalismo di memoria. Entrambi i formalismi di memoria sono rappresentati da derivate di ordine frazionario. Il mezzo è assunto anisotropico. Si trovano la trasformata di Laplace della funzione di Green in generale e la sua antitrasformata in casi particolari. 
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Caputo, Michele. 3-dimensional physically consistent diffusion in anisotropic media with memory. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 2, pp. 131-143. http://geodesic.mathdoc.fr/item/RLIN_1998_9_9_2_a7/

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