Speaker
Description
Fractional differential equations are an effective mathematical tool
to study anomalous diffusion phenomena, which are structurally
presented in real-life models such as oil pollution, tumor growth,
oxygen transport through capillaries to tissues, heat conduction in
living tissues. The feature of these anomalies in
diffusion/transport processes is that the mean square displacement
of the diffusing species $\langle (\Delta x)^{2}\rangle$ scales as a
nonlinear power law in time, i.e. $\langle (\Delta
x)^{2}\rangle\sim t^{\nu},$ $\nu>0$. For a subdiffusive process, the
value of $\nu$ belongs to $(0,1),$ while for normal diffusion
$\nu=1$, and for superdiffusive process, we have $\nu>1$. Appealing
to the fractional differential equations, this subdiffusion order
$\nu$ appears as the order of the leading fractional derivative in
time, $\mathbf{D}_{t}^{\nu}$, in the corresponding one- or
multi-term fractional differential operator $\mathbf{D}_{t}$ with
(in general) variable coefficients. A particular case of this
operator reads
$$
\mathbf{D}_{t}=\rho(x,t)\mathbf{D}_{t}^{\nu}+\sum_{i=1}^{M}\rho_{i}(x,t)\mathbf{D}_{t}^{\nu_i},\quad
\rho(x,t)>0,\quad 0\leq\nu_1<\nu_2<...<\nu_{M}<\nu<1.
$$
However, sometimes a value of the subdiffusion order (the order of
the fractional derivative in time) is not given a priori. In this
talk, we discuss a very effective (in practice) technique to
reconstruct the orders of fractional derivatives, coefficients in
the fractional operator $\mathbf{D}{t}$ and the order
$\gamma\in(0,1)$ of a singularity in the memory kernel
$\mathcal{K}=t^{-\gamma}\mathcal{K}{0}(t)$ in a semilinear
subdiffusion equation
$$
\mathbf{D}_{t}u-\mathcal{L}_{1}u-\int_{0}^{t}\mathcal{K}(t-\tau)\mathcal{L}_{2}u(x,\tau)d\tau=g(x,t,u),
$$
where $\mathcal{K}{0}$, $g$ are given functions,
$\mathcal{L}{1}$ and $\mathcal{L}{2}$ are uniform elliptic
operators of the second order with time and space depending
coefficients. It is worth noting that, the particular case of this
equation models the oxygen transport through capillaries to tissue.
In order to recognize these scalar parameters, we analyze the
corresponding inverse problem with additional local/nonlocal
measurement $\psi(t)$ for small time interval. Collecting Tikhonov regularization scheme and
quasioptimality approach, we describe a computational algorithm to
reconstruct these parameters in the case of discrete noisy
observation $\psi{\delta}(t_{i}),$ $i=1,...,K.$
