Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation

Xiaohua Jing; Masahiro Yamamoto

doi:10.3934/ipi.2022019

Article Contents

2022, Volume 16, Issue 5: 1199-1217. Doi: 10.3934/ipi.2022019

This issue Previous Article Using the Navier-Cauchy equation for motion estimation in dynamic imaging Next Article Reconstruction of singular and degenerate inclusions in Calderón's problem

Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation

Xiaohua Jing^1, , and
Masahiro Yamamoto^2,3,

1.
School of Sciences, Chang'an University, Xi'an, 710064, China
2.
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan
3.
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: Xiaohua Jing
^*Corresponding author: Xiaohua Jing

Received: December 2021

Revised: February 2022

Early access: April 2022

Published: October 2022

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Abstract

We consider two kinds of inverse problems on determining multiple parameters simultaneously for one-dimensional time-fractional diffusion-wave equations with derivative order $ \alpha \in (0, 2) $. Based on the analysis of the poles of Laplace transformed data and a transformation formula, we first prove the uniqueness in identifying multiple parameters, including the order of the derivative in time, a spatially varying potential, initial values, and Robin coefficients simultaneously from boundary measurement data, provided that no eigenmodes are zero. Our main results show that the uniqueness of four kinds of parameters holds simultaneously by such observation for the time-fractional diffusion-wave model where unknown orders $ \alpha $ vary order (0, 2) including 1, restricted to neither $ \alpha \in (0, 1] $ nor $ \alpha \in (1, 2) $. Furthermore, for another formulation of the fractional diffusion-wave equation with input source term in place of the initial value, we can also prove the simultaneous uniqueness of multiple parameters, including a spatially varying potential and Robin coefficients by means of the uniqueness result in the case of non-zero initial value and Duhamel's principle.

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

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