Diffusion with space memory modelled with distributed order space fractional differential equations

M. Caputo


Distributed order fractional differential equations (Caputo, 1995, 2001; Bagley and Torvik, 2000a,b) were fi rst used in the time domain; they are here considered in the space domain and introduced in the constitutive equation of diffusion. The solution of the classic problems are obtained, with closed form formulae. In general, the Green functions act as low pass fi lters in the frequency domain. The major difference with the case when a single space fractional derivative is present in the constitutive equations of diffusion (Caputo and Plastino, 2002) is that the solutions found here are potentially more fl exible to represent more complex media (Caputo, 2001a). The difference between the space memory medium and that with the time memory is that the former is more fl exible to represent
local phenomena while the latter is more fl exible to represent variations in space. Concerning the boundary value
problem, the difference with the solution of the classic diffusion medium, in the case when a constant boundary pressure is assigned and in the medium the pressure is initially nil, is that one also needs to assign the fi rst order space derivative at the boundary.


distributed order;fractional order;differential equations;constitutive equations;diffusion;space fractional derivative

Full Text:



DOI: https://doi.org/10.4401/ag-3395
We use cookies to ensure that we give you the best experience on our website. If you continue to use this site we will assume that you are happy with it (Read more).

Published by INGV, Istituto Nazionale di Geofisica e Vulcanologia - ISSN: 2037-416X