Boumenir, Amin; Tuan, Vu Kim An inverse problem for the wave equation. (English) Zbl 1279.34020 J. Inverse Ill-Posed Probl. 19, No. 4-5, 573-592 (2011). Summary: In the first part of this article, we show that we can recover the coefficient \(q\) in the one-dimensional wave equation from a finite number of special lateral measurements. Moreover, if some estimates on the size of \(q\) are available, then \(q\) can be recovered from a single boundary measurement. In the second part we treat the multidimensional case and show how we can reconstruct the coefficient \(q\) from a sequence of boundary measurements taken at one point only. Cited in 7 Documents MSC: 34A55 Inverse problems involving ordinary differential equations 34B24 Sturm-Liouville theory 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 35L05 Wave equation 35R30 Inverse problems for PDEs 42C15 General harmonic expansions, frames Keywords:inverse wave equation; inverse spectral theory; spectral estimation; initial-to-boundary map PDFBibTeX XMLCite \textit{A. Boumenir} and \textit{V. K. Tuan}, J. Inverse Ill-Posed Probl. 19, No. 4--5, 573--592 (2011; Zbl 1279.34020) Full Text: DOI References: [1] Avdonin S. A., Control Cybernet. 25 pp 429– (1996) [2] DOI: 10.1515/jiip.1997.5.4.309 · Zbl 0886.35166 · doi:10.1515/jiip.1997.5.4.309 [3] DOI: 10.3934/ipi.2010.4.1 · Zbl 1193.93071 · doi:10.3934/ipi.2010.4.1 [4] DOI: 10.1515/156939405775201718 · Zbl 1095.35061 · doi:10.1515/156939405775201718 [5] DOI: 10.1080/01630560903498979 · Zbl 1185.35320 · doi:10.1080/01630560903498979 [6] DOI: 10.1090/S0002-9939-2010-10297-6 · Zbl 1202.35344 · doi:10.1090/S0002-9939-2010-10297-6 [7] DOI: 10.1080/01630560903574993 · Zbl 1192.35186 · doi:10.1080/01630560903574993 [8] DOI: 10.1109/29.56027 · Zbl 0706.62094 · doi:10.1109/29.56027 [9] Isakov V., Contemp. Math. 268 pp 191– (2000) [10] DOI: 10.1070/RM1964v019n02ABEH001145 · doi:10.1070/RM1964v019n02ABEH001145 [11] DOI: 10.1137/1028003 · Zbl 0589.34024 · doi:10.1137/1028003 [12] DOI: 10.1088/0266-5611/11/2/013 · Zbl 0822.35154 · doi:10.1088/0266-5611/11/2/013 [13] Yamamoto M., J. Math. Kyoto Univ. 36 pp 825– (1996) [14] DOI: 10.1016/S0021-7824(99)80010-5 · Zbl 0923.35200 · doi:10.1016/S0021-7824(99)80010-5 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.