Abstract
An attempt is made to identify the orders of the fractional derivatives in a simple anomalous diffusion model, starting from real data. We consider experimental data taken at the Columbus Air Force Base in Mississippi. Using as a model a one-dimensional fractional diffusion equation in both space and time, we fit the data by choosing several values of the fractional orders and computing the infinite-norm “errors”, representing the discrepancy between the numerical solution to the model equation and the experimental data. Data were also filtered before being used, to see possible improvements. The minimal discrepancy is attained correspondingly to a fractional order in time around 0 . 6 and a fractional order in space near 2. These results may describe well the memory properties of the porous medium that can be observed.
Highlights
We provide some justification of the widespread use of simple linear parabolic fractional partial differential equations to model a number of physical phenomena, such as, for instance, diffusion of fluids in porous media [1] and in particular fluid flow of tracers through porous media
The aim of this work is that of identifying in a practical way the time and space orders of the fractional derivatives in a certain anomalous diffusion model, on the basis of the knowledge of real data
We used as a model a one-dimensional fractional diffusion equation in both, space and time, and fit the data by testing several possible, reasonable values for the fractional orders
Summary
We provide some justification of the widespread use of simple linear parabolic fractional partial differential equations (fPDEs) to model a number of physical phenomena, such as, for instance, diffusion of fluids in porous media [1] and in particular fluid flow of tracers through porous media. The experimental setup was designed aiming at determining the memory properties of the observed dynamical behavior The latter could be considered as an anomalous diffusion, and was well reproduced by a suitable fPDE modeling of the constitutive laws involving pressure gradient–flux and pressure–density variations. A code was written to compute Un, by minimizing, at each time step, the ∞ norms (with respect to space) of the discrepancies between the experimental and the numerically computed values obtained by solving (numerically) the aforementioned anomalous diffusion equation. Recently, in [2], experimental results have been obtained to estimate the physical parameters governing the dynamical behavior of the flow of tracers through a certain given porous medium, in particular to observe the aforementioned memory effects described by parameters such as, for instance, the order of the fractional derivative with respect to time. The function to minimize is the error, i.e., the discrepancy between the numerically computed value, Un, and the value of u(x, t) obtained by using the real experimental data, Vn, that is Un − Vn ∞
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