Dipartimento di Scienze di Base e Applicate per l'Ingegneria

PhD Course:  Inverse problems and time-fractional partial differential equations
Masahiro Yamamoto (The University of Tokyo)
Duration: 16 hours
Period: September/October 2022, Aula 1B1
 14, 20, 22, 27 , 28 September  (10:00 - 12:00)
4, 5, 7 October  (10:00 - 12:00)

Abstract: We consider an initial boundary value problem for time-fractional diffusion-wave equation (in the following we refer to it as system (*)).

The lectures aim at self-contained concise explanations for the fundamental theory for such problems and mathematical 
analysis of inverse problems.

The idea of fractional derivatives dates back to Leibniz and there have been many works including by Abel, Riemann, Liouville. Now the system (*) is widely recognized as more feasible model for various phenomena such as anomalous diffusion in heterogeneous media, where the anomaly cannot be well interpreted by the classical advection. diffusion equation and the conventional models often provide wrong simulation results. Thus we have to exploit more relevant models because the issues are serious, for example, for the protection of the environments , and the mathematical researches should support such practical applications.
For researches on inverse problems, we need also mathematical analyses for (*). Usually in practice, we are not a priori given coefficients and other quantities in (*). The inverse problems are concerned with parameter identification, and are essential for more accurate prediction or simulations of anomalous diffusion. Therefore mathematical researches should be done for both foundations of direct problems and applications to inverse problems for (*).

Inverse problems and time-fractional partial differential equations by M. Yamamoto (The Univ. Tokyo)
I plan the following contents under possible changes.
 
September 14

1. Introduction: motivation for time-fractional diffusion equations
2. Elementary explanations of time-fractional derivatives
3. Initial-boundary value problem for simple time fractional ordinary differential equations

September 20

1. Initial-boundary value problems for time-fractional diffusion-wave equations: solution  by the Fourier method
2. Some qualitative studies of solutions

September 22

1. Qualitative properties (e.g., asymptotics)
2. Several inverse problems

September 27

1. Issues on time-fractional derivatives: motivations for operator theoretic approach
2. Formulation of fractional derivatives in Sobolev spaces of positive orders
3. Extensions to Sobolev spaces of negative orders

September 28

1. Important properties of the extended time-derivatives

(Laplace transform, coercivity, successive differentiation)

2. Time-fractional ordinary differential equations: solution formulae in Sobolev spaces

October 4

1. Initial-boundary value problems and nonlinear equations
2. Comparison principles and applications

October 5

1. Inverse problems of determining orders of fractional derivatives
2. Inverse problems of determination of source terms

October 7

1. Inverse problems of determining initial values
2. Backward problem in time

Prospects for future researches