Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations

Masahiro Yamamoto

doi:10.3934/mcrf.2022017

Article Contents

2023, Volume 13, Issue 2: 833-851. Doi: 10.3934/mcrf.2022017

This issue Previous Article Averaged turnpike property for differential equations with random constant coefficients Next Article

Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations

Masahiro Yamamoto^,

Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan, Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania, Correspondence Member of Accademia Peloritana dei Pericolanti, Palazzo Università, Piazza S. Pugliatti 1 98122 Messina, Italy

^* Corresponding author: Masahiro Yamamoto
^* Corresponding author: Masahiro Yamamoto

Received: April 2021

Revised: November 2021

Early access: April 2022

Published: June 2023

The author was supported by JSPS grant 20H00117 and NSFC grants 11771270, 91730303.

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Abstract

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $ \partial_t^{\alpha} u(x, t) = -Au(x, t) $, where $ -A = \sum_{i, j = 1}^d \partial_i(a_{ij}(x) \partial_j) + \sum_{j = 1}^d b_j(x) \partial_j + c(x) $. We establish the uniqueness for an inverse problem of determining an order $ \alpha $ of fractional derivatives by data $ u(x_0, t) $ for $ 0<t<T $ at one point $ x_0 $ in a spatial domain $ \Omega $. The uniqueness holds even under assumption that $ \Omega $ and $ A $ are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.

Keywords:

Fractional advection-diffusion equation,
uniqueness,
fractional order,
generalized eigenspace,
solution representation.

Mathematics Subject Classification: Primary: 35R30, 35R11.

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