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

Abstract

<p style='text-indent:20px;'>We consider two kinds of inverse problems on determining multiple parameters simultaneously for one-dimensional time-fractional diffusion-wave equations with derivative order <inline-formula><tex-math id="M1">\begin{document}$ \alpha \in (0, 2) $\end{document}</tex-math></inline-formula>. Based on the analysis of the poles of Laplace transformed data and a transformation formula, we first prove the uniqueness in identifying multiple parameters, including the order of the derivative in time, a spatially varying potential, initial values, and Robin coefficients simultaneously from boundary measurement data, provided that no eigenmodes are zero. Our main results show that the uniqueness of four kinds of parameters holds simultaneously by such observation for the time-fractional diffusion-wave model where unknown orders <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> vary order (0, 2) including 1, restricted to neither <inline-formula><tex-math id="M3">\begin{document}$ \alpha \in (0, 1] $\end{document}</tex-math></inline-formula> nor <inline-formula><tex-math id="M4">\begin{document}$ \alpha \in (1, 2) $\end{document}</tex-math></inline-formula>. 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.</p>