Inverse moving source problem for time-fractional evolution equations: Determination of profiles

Abstract

This paper is concerned with the determination of moving source profile functions arising from time-fractional diffusion(-wave) equations, if the sources are supposed to move along given straight lines. If the time-fractional order satisfies 0 < α ≤ 1, we prove uniqueness in recovering a single moving source profile from suitably chosen volume observations over a finite time interval. If 1 < α ≤ 2, unique identification of two source profiles with distinct moving directions is verified.

Numerical schemes for reconstructing profiles of moving sources in (time-fractional) evolution equations

“…Recently, it has been recognized that (time-fractional) evolution equations with orders 0 < α < 2 actually share several common properties, which can be applied to the qualitative analysis of some inverse problems. Besides the time-analyticity asserted in [24], the weak vanishing property was established in [7,17] for 0 < α < 1 and 1 < α < 2 respectively, which resembles the classical unique continuation principle for parabolic equations. As direct applications, this property was employed to prove the uniqueness of an inverse x-source problem in [7] and that of Problem 1.1 in [17].…”

“…As a continuation of the theoretical counterpart studied in [17], this article is concerned with the establishment of numerical reconstruction schemes for the following inverse moving source problem regarding (1.1)-(1.2). Problem 1.1 (Inverse moving source problem) Let u be the solution to (1.1)-(1.2), and ω ⊂ Ω be a suitably chosen nonempty subdomain of Ω.…”

Simultaneous uniqueness for an inverse problem in a time-fractional diffusion equation

“…We apply an argument in Liu, Hu and Yamamoto [10], which relies on the analysis of the poles of Laplace transformed data.…”

Unique continuation property for multi-terms time fractional diffusion equations

“…For the fractional anomalous diffusion equation with single term time fractional derivative, it was proved in [7] for the one space dimensional case and it was also proved in [6] for the higher space dimensional case. Even a further generalization of [6] has been done in [10] for the case that the order of the time fractional derivative is in (0, 2) and the elliptic part of the equation is the minus Laplacian. Furthermore, the weak UCP was used to solve the uniqueness of some inverse source problems ( [6], [10], [11] and the references there in).…”

“…Even a further generalization of [6] has been done in [10] for the case that the order of the time fractional derivative is in (0, 2) and the elliptic part of the equation is the minus Laplacian. Furthermore, the weak UCP was used to solve the uniqueness of some inverse source problems ( [6], [10], [11] and the references there in). This already revealed the importance of the classical UCP in the studies of many inverse problems for the anomalous diffusion equations.…”