Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918534606487552 |
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| author | Yu, Kai |
| author_facet | Yu, Kai |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01812 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements Yu, Kai Analysis of PDEs 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. |
| title | Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2606.01812 |