On a multidimensional free boundary problem governed by anomalous diffusion: analytical and numerical study

Nataliya Vasylyeva
Institute of Mathematics NAS of Ukraine
Wednesday 23rd March 2016
Aula Seminari
Other the past few decades an intensive effort has been put into developing theoretical models for systems with diffusive motion that can not be modelled as standard Brownian motion. The signature of this anomalous di®usion is that the mean square displacement of the diffusing species scales as a nonlinear power law in time, i.e. behaves as t to the power alpha. If alpha belongs to (0,1), this is referred to as subdi®usion. In recent years the additional motivation for these studies has been stimulated by experimental measurements of subdi®usion in porous media, glass forming materials, biological media. The review paper by Klafter et al. provides numerous references to physical phenomena in which anomalous diffusion occurs. Here we analyze anomalous di®usion version of the quasistationary Stefan problem (the fractional Hele-Shaw problem) in the multidimensional case ¬Omega(t) in R^n, n greater or equal to 2. This free boundary problem in the case of zero surface tension of the moving boundary is a mathematical model of a solute drug release from a polymer matrix (n = 1; 3). First we represent results related with a classical solvability of this moving boundary problem. Second we discuss a numerical technique which has been applied to obtain numerical simulations of the solutions. The relevant mathematical models of drug release from a polymeric matrix are powerful tools in studies of controlled-release drug system. Therefore, the investigations of fractional calculus in moving boundary problems would be of great interest to both theoretical and experimental studies in the future. contact: chritian.vergara@polimi.it Brief CV: She has obtained her PhD degree in Mathematics in 2003 on free boundary problems with singular initial data. She is working as Senior Researcher at the Institute of Applied Mathematics and Mechanics of NAS of Ukraine. Her main research activities are in the field of free boundary problems, in the field of boundary value problems for elliptic and parabolic equations in domains with nonsmooth boundaries; in the field of fractional calculus and its application to boundary value problems. She is author more than 38 publications. She has been the local coordinators of Marie Curie IRSES project (“EUMLS”) in FP7 and she is currently the local coordinator of Marie Curie RISE project (“AMMODIT”) in Horizon 2020. She is a member of Editorial board in Universal Journal of Applied Mathematics”;“American Journal of Computational and Applied Mathematics”, “Fractional Differential Calculus”.