Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior. (29th December 2022)

• Record Type: Journal Article
• Authors: Liu, Yikan; Yamamoto, Masahiro
• Is Part Of: Inverse problems. Volume 39:Number 2(2023)
• Publication Date: 2022-12-29
• Subjects: multi-term time-fractional diffusion equation -- parameter inverse problem -- uniqueness -- short-time asymptotic behavior; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/acab7a
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently investigated in literature. Unlike all the existing results, in this article we prove the uniqueness of orders and parameters (up to a multiplier for the latter) only by principal terms of asymptotic expansions of solutions near t = 0 at a single spatial point. Moreover, we discover special conditions on unknown initial values or source terms for the coincidence of observation data. As a byproduct, we can even conclude the uniqueness for initial values or source terms by the same data. The proof relies on the asymptotic expansion after taking the Laplace transform and the completeness of generalized eigenfunctions.