ENG 
ESP Session S08 - Inverse Problems and Applications
Thursday, July 15, 17:50 ~ 18:20 UTC-3
Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem
Silvia Seminara
Facultad de Ingeniería - Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
The motion of microscopic particles in a fluid has been investigated for a long time. The "Brownian motion'' consists of random displacements of the suspended particles within a medium. The molecules of the fluid, much smaller than the solid particles, knock them incessantly, both driving and damping their movement; macroscopically, those knockings give rise to the "fluid viscosity''.
The traditional mathematical model for this diffusion phenomenon is based on Einstein's theory by which the mean square displacement of diffusion particles is proportional to time. This idea leads to the classical diffusion equation, $$ u'_t({\textbf{x}},t)-k\nabla^2 u({\textbf{x}},t)=0,$$ where $u({\textbf{x}},t)$ is the quantity of particles per unit volume at position ${\textbf{x}}$ and time $t$, and $k$ is a diffusion coefficient. If there is also a source of additional incoming particles, the equation becomes non-homogeneous: $$u'_t({\textbf{x}},t)-k\nabla^2 u({\textbf{x}},t)=s({\textbf{x}},t).$$
But there are experimental results - like anomalous diffusion of particles in porous or fractal media, biological media, turbulent plasmas, polymers, etc. - that show that, in some cases, the mean square displacement of particles must be considered to be proportional not to the time, but to a fractional power of time, to fit the empirical data. This fractional order may be less than unity (subdiffusion) or greater than one (superdiffusion).
Fractional differential equations have been proposed to model this anomalous diffusion phenomenon: $${\cal{D}}_t^\alpha u({\textbf{x}},t)-k\nabla^2 u({\textbf{x}},t)=s({\textbf{x}},t),$$ where $\alpha$ is a not integer order of derivation with respect to time; $0<\alpha<1$ for subdiffusion, $\alpha>1$ for superdiffusion.
There are several definitions of ${\cal{D}}_t^\alpha$ (Riemann-Liouville's, Caputo's, Atangana-Baleanu's, Caputo-Fabrizio's, etc.), all of them involving an integral operator which takes account of the "past history'' of the function.
In this work, we have chosen the Caputo-Fabrizio fractional derivative to describe a 1-dimensional model of subdiffusion of the form $${\cal{D}}_t^\alpha u(x,t)-k\frac{\partial^2}{\partial x^2} u(x,t)=s(x)h(t),$$ where $x\in (0,1)$ and $t\in (0,T)$ and the source is supposed to be a separable function of variables $x$ and $t$.
We are interested in solving the inverse problem that consists in finding $s(x)$ and $u(x,t)$, for a known $h(t)$ and additional data: measurements $u(x_i,T), \, x_i\in (0,1)$, for $i=1, \cdots, N$.
We obtained an approximate solution by separating variables and solving the resulting fractional differential equation. We present some numerical examples that show the good performance of the proposed scheme and state some conclusions.
Joint work with María Inés Troparevsky (Universidad de Buenos Aires), Marcela Fabio (Universidad de San Martín) and Guillermo La Mura (Universidad de San Martín).