Speaker : Masahiro Yamamoto, The University of Tokyo, Zonguldak B¨ulent Ecevit University

Time : 15:00 - 16:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

YouTube :   https://youtu.be/R8B-NYBuBt0

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: inverse problems, time-fractional differential equations, uniqueness and stability

Abstract: One should be concerned with many kinds of the diffusion of substances in heterogeneous media, and when one considers diffusion of contaminants such as Caesium-137, such studies may be very serious issues for public safety. In the diffusion in heterogeneous media, it is often observed that profiles of density of substances indicate large deviation
from the classical diffusion profiles. Therefore, more suitable model equations are demanded and time-fractional diffusion-wave equations is one hopeful model equation and call great attention.

Anomalous diffusion is characterized by slow diffusion in some sense and long-tail profiles, and for fractional equations we can prove slow decay for large time, less smoothing property of distribution, etc. Thus the time-fractional diffusion-wave equation is very probable model

d_t^α u(x, t) − Au(x, t) = F (x, t), x ∈ Ω, 0 < t < T

and will study more gerenal forms. Here Ω is a bounded domain in R^d, d ∈ N, −A is a uniform elliptic operator of the second order. Moreover d_t^α with α ∈ (0, 2) \ {1}, is a Caputo type of fractional derivative.

Introducing time-fractional diffusion equations for modelling actual anomalous diffusion, we should first identify parameters governing diffusion. As such parameters, we refer
for example to the orders of time-derivative, source terms. This is nothing but inverse problems.

The main purpose of this talk is to study several kinds of inverse problems for time-fractional diffusion-wave equations (1) and present results on the uniqueness and the sta-
bility. We mainly study the following kinds of inverse problems:

(I) Determination of orders α and related quantities.

(II) Determination of μ(t) or f(x) in the case where a source F (x, t) is modelled in a form F (x, t) = μ(t)f (x).

(III) Backward problems in time: for fixed T > 0 determine
       u(·, 0) by u(·, T ) if 0 < α < 1,
       u(·, 0), ∂_tu(·, 0)} by {u(·, T ), ∂tu(·, T )} if 1 < α < 2.

It is important: which kind of data should we adopt for the uniqueness for inverse problems? Adequate choices of data essentially depend on properties of solutions to initial boundary value problems for time-fractional diffusion-wave equations. Thus our studies for inverse problems should be grounded on the direct problems, and we will explain also necessary knowledge of direct problems

Biography:  Professor Masahiro Yamamoto received his Ph.D. from the University of Tokyo in March 1988. He has been a professor at the Graduate School of Mathematical Sciences of the University of Tokyo since April 2010, where he was previously a research associate from April 1985 to March 1990 and an associate professor from April 1990 to March 2010. Since 2024, he has also been a professor at Zonguldak B¨ulent Ecevit University in Turkey. His research topics include inverse problems for partial differential equations, fractional partial differential equations, and industrial mathematics. He has published almost 400 peer-reviewed papers, and received almost 8,000 citations from more than 3,000 publications (according to the Mathematical Reviews database of the American Mathematical Society

Bibliography