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arxiv: 1907.02430 · v1 · pith:YZJQ3EIInew · submitted 2019-07-04 · 🧮 math.AP

Unique determination of several coefficients in a fractional diffusion(-wave) equation by a single measurement

Yavar Kian , Zhiyuan Li , Yikan Liu , Masahiro Yamamoto This is my paper

Pith reviewed 2026-05-25 09:08 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse problemsfractional diffusion equationsunique determinationboundary measurementscoefficient recoveryfractional wave equationsNeumann data
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The pith

Several time-independent coefficients in fractional diffusion or wave equations are uniquely recovered from one Neumann boundary measurement with a suitable input.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a single Neumann measurement on part of the boundary, taken while a suitable input is applied on another part, suffices to recover multiple time-independent coefficients in fractional diffusion and wave equations posed on bounded domains. These coefficients include quantities such as the density of the medium and the velocity field of the moving quantities. A reader would care because the result reduces the number of experiments or sensors needed to identify parameters in models of anomalous diffusion and nonlocal wave propagation. The argument covers both standard and fractional-order time derivatives within a general class of coefficients. The uniqueness holds under the stated boundary restrictions without requiring full knowledge of the solution inside the domain.

Core claim

For fractional diffusion(-wave) equations on a bounded domain, several coefficients from a general class of time-independent coefficients are uniquely determined by a single Neumann boundary measurement, on some parts of the boundary, of the solution corresponding to a suitable boundary input located on some parts of the boundary.

What carries the argument

The single Neumann boundary measurement of the solution under a suitable boundary input, which encodes the effects of the unknown coefficients through the fractional time derivative and the spatial differential operators.

If this is right

  • The density of the medium is recoverable from the single measurement.
  • The velocity field associated with the transported quantities is recoverable from the single measurement.
  • The result applies equally to equations with ordinary time derivatives and to those with fractional time derivatives.
  • Unique recovery holds when data are restricted to proper subsets of the boundary rather than the entire boundary.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may reduce experimental cost in laboratory settings where only a few boundary sensors are feasible.
  • Similar single-measurement uniqueness could be tested for coefficients that vary mildly with time or for equations on unbounded domains.
  • The approach suggests examining whether the same boundary data can also recover nonlinear terms or source functions in related fractional models.

Load-bearing premise

The coefficients remain constant in time while the domain stays bounded and both the input and the measurement are confined to portions of the boundary.

What would settle it

Exhibit two different sets of time-independent coefficients that produce exactly the same single Neumann boundary measurement for the same boundary input.

read the original abstract

We consider the inverse problem of determining different type of information about a diffusion process, described by ordinary or fractional diffusion equations stated on a bounded domain, like the density of the medium or the velocity field associated with the moving quantities from a single boundary measurement. This properties will be associated with some general class of time independent coefficients that we recover from a single Neumann boundary measurement, on some parts of the boundary, of the solution of our diffusion equation with a suitable boundary input, located on some parts of the boundary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves uniqueness results for recovering multiple time-independent coefficients (including density and velocity fields) simultaneously in fractional diffusion and fractional wave equations on a bounded domain, using a single Neumann boundary measurement of the solution corresponding to one suitable boundary input, both located on unspecified parts of the boundary.

Significance. If the proofs are complete, the result strengthens the theory of inverse problems for nonlocal PDEs by showing that several coefficients can be recovered from minimal single-measurement data, which is a notable technical achievement compared to typical multi-measurement or infinite-data requirements in the literature.

major comments (2)
  1. [Abstract and §2–§3] The central uniqueness claim (stated in the abstract and likely formalized in the main theorems of §3–§5) depends on the input set Γ_in and measurement set Γ_m being chosen so that information from the interior coefficients propagates to the boundary. The repeated phrasing “on some parts of the boundary” provides no explicit open-set, positive-measure, or geometric-control condition on these subsets. Standard Carleman-estimate or unique-continuation arguments for fractional operators require at least an open portion of the boundary (or a microlocal condition) to close the estimates; without such a hypothesis the single-measurement map need not be injective even for smooth positive coefficients.
  2. [Main theorems (likely §4–§5)] The reduction from the fractional diffusion case to the wave case (or vice versa) via analytic continuation in time or Laplace transform is load-bearing for the “several coefficients” claim. The manuscript must verify that the same single pair (Γ_in, Γ_m) works uniformly for both equations without introducing extra assumptions on the coefficients that are not stated in the abstract.
minor comments (2)
  1. [§1–§2] Notation for the fractional order and the precise function spaces for the coefficients should be introduced earlier and used consistently.
  2. [Abstract] The abstract would benefit from a single sentence stating the precise regularity assumed on the unknown coefficients.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below. We agree that the boundary-set hypotheses require more explicit wording and will revise accordingly, but the proofs already rely on standard open-set assumptions that suffice for the unique-continuation arguments employed.

read point-by-point responses

  1. Referee: [Abstract and §2–§3] The central uniqueness claim (stated in the abstract and likely formalized in the main theorems of §3–§5) depends on the input set Γ_in and measurement set Γ_m being chosen so that information from the interior coefficients propagates to the boundary. The repeated phrasing “on some parts of the boundary” provides no explicit open-set, positive-measure, or geometric-control condition on these subsets. Standard Carleman-estimate or unique-continuation arguments for fractional operators require at least an open portion of the boundary (or a microlocal condition) to close the estimates; without such a hypothesis the single-measurement map need not be injective even for smooth positive coefficients.

    Authors: In the manuscript the sets Γ_in and Γ_m are nonempty open subsets of ∂Ω (with positive surface measure). This is the precise geometric hypothesis used to invoke the unique-continuation and Carleman-estimate results for the fractional operators (see the statements preceding Theorems 3.1 and 4.2 and the proofs in §4). The informal phrasing “some parts” is therefore understood in the standard sense of open boundary portions, but we acknowledge that the abstract and introductory paragraphs should state the condition explicitly. We will revise the abstract, §2, and the main theorem statements to read: “Γ_in and Γ_m are nonempty open subsets of ∂Ω.” No change to the proofs is required. revision: yes

  2. Referee: [Main theorems (likely §4–§5)] The reduction from the fractional diffusion case to the wave case (or vice versa) via analytic continuation in time or Laplace transform is load-bearing for the “several coefficients” claim. The manuscript must verify that the same single pair (Γ_in, Γ_m) works uniformly for both equations without introducing extra assumptions on the coefficients that are not stated in the abstract.

    Authors: Section 5 performs the reduction by the Laplace transform in the time variable. Because the transform acts only in time and the boundary input/measurement are time-independent in their spatial support, the same fixed pair (Γ_in, Γ_m) is used for both the fractional diffusion and wave equations. The coefficient assumptions (time-independent, belonging to the stated Sobolev classes) are identical for both equations and are not strengthened by the transform. Analytic continuation in the Laplace parameter then transfers uniqueness directly, without additional geometric or coefficient hypotheses. The argument is therefore uniform and already contained in the manuscript. revision: no

Circularity Check

0 steps flagged

No circularity: uniqueness theorem derived from independent PDE analysis

full rationale

The paper establishes a mathematical uniqueness result for recovering several time-independent coefficients in fractional diffusion(-wave) equations from a single Neumann boundary measurement. The derivation relies on standard analytic techniques (e.g., Carleman estimates or unique continuation) applied to the PDE system on a bounded domain, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations that collapse the claim to its inputs. The abstract and setup describe a direct inverse-problem theorem whose validity is independent of the specific measurement subsets once the stated geometric conditions are met; no step renames a known empirical pattern or imports an ansatz via prior self-work as the sole justification. This is a self-contained existence/uniqueness proof, not a statistical prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Extracted solely from the abstract; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Coefficients are time-independent
    Explicitly stated in the abstract as 'time independent coefficients'
  • domain assumption Equations stated on a bounded domain
    Stated in the abstract as 'stated on a bounded domain'

pith-pipeline@v0.9.0 · 5615 in / 1299 out tokens · 36986 ms · 2026-05-25T09:08:32.059355+00:00 · methodology

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