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

Abstract

Distributed order fractional differential equations (Caputo, 1995, 2001; Bagley and Torvik, 2000a,b) were first 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 filters 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 flexible 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 flexible to represent local phenomena while the latter is more flexible 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 first order space derivative at the boundary.