Simultaneous determination of initial value and source term for time-fractional wave-diffusion equations

TL;DR

This paper addresses the inverse problem of simultaneously determining the initial condition and source term in time-fractional wave-diffusion equations using boundary data, establishing uniqueness under certain conditions.

Contribution

It proves the uniqueness of recovering initial values and source terms in time-fractional diffusion-wave equations from boundary measurements, considering the fractional order constraints.

Findings

Uniqueness of simultaneous determination of initial data and source term.

Conditions on fractional order for the uniqueness result.

Use of Mittag-Leffler functions' asymptotic behavior in proofs.

Abstract

We consider initial boundary value problems for time fractional diffusion-wave equations: $$ d_t^{\alpha} u = -Au + \mu(t)f(x) $$ in a bounded domain where $\mu(t)f(x)$ describes a source and $\alpha \in (0,1) \cup (1,2)$, and $-A$ is a symmetric ellitpic operator with repect to the spatial variable $x$. We assume that $\mu(t) = 0$ for $t > T$:some time and choose $T_2>T_1>T$. We prove the uniqueness in simultaneously determining $f$ in $\Omega$, $\mu$ in $(0,T)$, and initial values of $u$ by data $u\vert_{\omega\times (T_1,T_2)}$, provided that the order $\alpha$ does not belong to a countably infinite set in $(0,1) \cup (1,2)$ which is characterized by $\mu$. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.