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Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements

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Pith's one-line read If two coefficient pairs yield the same DtN map, the fractional damping coefficient and potential coincide almost everywhere.

desk verdict This paper proves uniqueness of the fractional damping coefficient and potential from the DtN map for a time-fractional damped wave equation by adapting beam solutions to the Caputo kernel. read the letter →

arxiv 2606.01812 v1 pith:55K557LF submitted 2026-06-01 math.AP

Kai Yu This is my paper
classification math.AP
keywords inversecoefficientproblemtime-fractionalwaveequationDirichlet-to-NeumannmapuniquenessCaputoderivativeboundarymeasurementsdampedfractionaldamping
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the Dirichlet-to-Neumann map uniquely determines the fractional damping coefficient and the zeroth-order potential in a time-fractional damped wave equation on a finite time interval. The result follows from a convolution integral identity that cancels unknown boundary terms and from high-frequency beam solutions whose asymptotics isolate the coefficients. A reader would care because recovering these coefficients from boundary data supplies the mathematical basis for identifying material properties in systems with memory effects such as viscoelasticity.

What carries the argument

high-frequency beam solutions adapted to the singular kernel of the Caputo derivative together with asymptotic analysis of fractional convolution terms

What would settle it

The existence of two distinct pairs of coefficients that generate identical Dirichlet-to-Neumann maps for the same domain and time interval.

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Extended reading notes

Core claim

The paper shows that two pairs of coefficients consisting of the fractional damping term and the potential produce the same Dirichlet-to-Neumann map if and only if the coefficients agree almost everywhere. The argument relies on first establishing well-posedness of the forward problem, then deriving a convolution-type identity from the equation, constructing high-frequency solutions adapted to the Caputo kernel, and performing asymptotic analysis on the resulting terms to recover the coefficients pointwise.

Load-bearing premise

The construction of high-frequency beam solutions adapted to the singular kernel of the Caputo derivative together with the detailed asymptotic analysis suffice to eliminate unknown boundary traces and recover the coefficients.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes well-posedness of the forward problem for a time-fractional damped wave equation with suitable boundary lifting and proves a uniqueness theorem: if two pairs of coefficients (fractional damping coefficient and zeroth-order potential) produce the same Dirichlet-to-Neumann map on a finite time interval, then the coefficients coincide almost everywhere. The argument proceeds via a convolution-type integral identity that removes unknown boundary traces, followed by construction of high-frequency beam solutions adapted to the Caputo kernel and asymptotic analysis of the resulting fractional convolutions.

Significance. If the result holds, it provides a rigorous extension of classical uniqueness results for hyperbolic inverse problems to a fractional-order model incorporating damping and memory effects. This supplies a theoretical foundation for applications involving viscoelasticity and anomalous diffusion. The technical adaptation of beam solutions to the singular Caputo kernel, together with the explicit asymptotic treatment of fractional convolutions, constitutes a reusable contribution to the literature on inverse problems for non-local PDEs.

minor comments (3)
  1. [Abstract and §2] The abstract refers to 'boundary data with sufficient regularity to admit a suitable lifting' without stating the precise Sobolev or Hölder spaces; the well-posedness section should list the exact function spaces for the Dirichlet data and the solution to make the lifting construction reproducible.
  2. [Introduction] Notation for the Caputo derivative, the two unknown coefficients, and the DtN map is introduced gradually; a consolidated notation table or paragraph at the end of the introduction would improve readability for readers unfamiliar with fractional inverse problems.
  3. [§4] The asymptotic analysis of the fractional convolution terms in the beam-solution step relies on several integration-by-parts identities; a short appendix collecting these identities with explicit remainder estimates would strengthen the exposition without lengthening the main argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its significance for extending uniqueness results to fractional-order models, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

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No significant circularity

full rationale

The paper establishes uniqueness of two coefficients from the DtN map via an explicit proof strategy: a convolution integral identity eliminating unknown boundary traces, followed by construction of high-frequency beam solutions adapted to the Caputo kernel and asymptotic analysis of fractional convolutions. No quoted step reduces by definition to its own inputs, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation chain. The argument is presented as an independent derivation extending classical hyperbolic techniques, with the central claim self-contained against the stated forward well-posedness and recovery steps.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the argument relies on standard functional-analytic well-posedness and asymptotic techniques whose precise statements are not given.

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Cite this review

Pith. "Pith review of Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements." pith.science (2026). https://pith.science/paper/55K557LF

@misc{pith2026260601812,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55K557LF}},
  note         = {Machine review of arXiv:2606.01812}
}
read the original abstract

This paper studies an inverse coefficient problem for a time-fractional damped wave equation on a finite time interval. The aim is to determine two spatially varying coefficients, namely the fractional damping coefficient and the zeroth-order potential, from the associated Dirichlet-to-Neumann (DtN) map. We first prove the well-posedness of the forward problem for boundary data with sufficient regularity to admit a suitable lifting. The main result is a uniqueness theorem showing that if two coefficient pairs give rise to the same DtN map, then the corresponding coefficients coincide almost everywhere in the domain. The proof is based on a convolution-type integral identity that eliminates the unknown boundary traces, the construction of high-frequency beam solutions adapted to the singular kernel of the Caputo derivative, and a detailed asymptotic analysis of the resulting fractional convolution terms. This result extends classical uniqueness results for hyperbolic inverse problems to a fractional-order model with damping and provides a theoretical basis for related applications involving memory and viscoelastic effects.

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