Uniqueness in determining fractional orders of derivatives and initial values

Abstract

For initial boundary value problems of time-fractional diffusion-wave equations with the zero Dirichlet boundary value, we consider an inverse problem of determining orders of the fractional derivatives and initial values by three kinds of measurement data on a time interval: (i) solution in a subdomain (ii) Neumann data on a subboundary (iii) spatial averaged values. We prove that the order and initial values are uniquely determined by the above data if the solutions do not identically vanish. Our main results show that the uniqueness for the fractional order holds even if the coefficients of the fractional diffusion-wave equations and source terms within some class are unknown. Moreover we consider pointwise data for the uniqueness in the inverse problem. The proof is based on the eigenfunction expansions and the asymptotic expansions of the Mittag-Leffler functions for large time.

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

“…On the other hand, since the advection term d j=1 b j (x)∂ j u is of lower-order and does not drastically change the structure of the equation, for non-symmetric A, we can naturally expect a similar uniqueness result to [4], [12], [32], [33]. The main purpose of this article is to prove that such a conjecture is correct for A given by (1.1).…”

“…The proof relies on the asymptotic behavior of u and u as t → ∞, which is an idea similar to Sakamoto and Yamamoto [28], Yamamoto [33], but by the non-symmetry of A and A, we cannot make use of the eigenfunction expansions of tne solutions u and u themselves. Alternatively we derive asymptotic expansions of eigenprojections of u and u in view of the completeness in L 2 (Ω) of generalized eigenfunctions for each of A and A.…”

“…and Kian [12], Krasnoschok, Pereverzyev, Siryk and Vasylyeva [15], Li, Zhang, Jia and Yamamoto [17], Li and Yamamoto [20], Tatar, Tinaztepe and Ulusoy [29], Tatar and Ulusoy [30], Yamamoto [32], [33], Yu, Jiang and Qi [34], for example. See Li, Liu and Yamamoto [19] as a survey.…”

“…We can prove the uniqueness by specified data of solution of even if the coefficients of A are unknown, and we refer to [5] as early work, and see [12], [33] as related articles.…”

“…The main purpose of this article is to prove that such a conjecture is correct for A given by (1.1). Moreover, suggested by [5], [12] and [33], the order α can be uniquely determined independently of the operator A and the domain Ω.…”