Inversion and prediction problems in anomalous diffusion

2017 Fiscal Year Final Research Report

• Project/Area Number: 16H06712
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Mathematical analysis
• Research Institution: The University of Tokyo
• Principal Investigator: Li Zhiyuan 東京大学, 大学院数理科学研究科, 特任研究員 (00782450)
• Research Collaborator: YAMAMOTO Masahiro 東京大学, 大学院数理科学研究科, 教授 ; LUCHKO Yuri Beuth Technical University of Applied Sciences Berlin, 教授 ; JIANG Dai jun 華中师范大学, 数学統計学院, 准教授 ; LIU Yikan 東京大学, 大学院数理科学研究科, 助教
• Project Period (FY): 2016-08-26 – 2018-03-31
• Keywords: 拡散方程式 / 逆問題

Outline of Final Research Achievements

The diffusion equation with Caputo derivative was discussed. The Caputo derivative is inherently nonlocal in time with history dependence, which makes the crucial differences between fractional models and classical models. What about the unique continuation (UC)? There is not affirmative answer to this problem except for some special cases. By using Theta function method and Laplace transform argument, we gave a classical type unique continuation, say, the vanishment of a solution to a the fractional diffusion equation in an open subset implies its vanishment in the whole domain provided the solution vanishes on the whole boundary.
We also considered an inverse problem in determining the fractional order. By exploiting the integral equation of the solution u to the our problem, and carrying out the inversion Laplace transforms, we verified the Lipschitz continuous dependency of the fractional order with respect to the overposed data.

Free Research Field

偏微分方程式の逆問題