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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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1569-3945
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Volume 25, Issue 1

Issues

Reconstruction of the refractive index from transmission eigenvalues for spherically stratified media

Xiao-Chuan Xu
  • Corresponding author
  • Department of Applied Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P. R. China
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/ Chuan-Fu Yang
  • Department of Applied Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P. R. China
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Published Online: 2015-11-28 | DOI: https://doi.org/10.1515/jiip-2015-0081

Abstract

This work deals with the interior transmission eigenvalue problem for a spherically stratified medium supported in {x:|x|=r1}, which can be formulated as y′′+k2η(r)y=0 with boundary conditions y(0)=0=y(1)(sink/k)-y(1)cosk. Based on Rundell and Sacks’s idea in [11], the inverse problem of recovering the function η(r) from the spectrum of the problem in the case 01η(r)𝑑r=1 is studied. In general, it is known that for the unique determination of η(r), it is necessary to specify along with the spectrum also a constant γ which is the ratio of the characteristic determinant of the problem. We provide several algorithms for recovering η(r) from the spectrum. In the first algorithm, it is supposed that the constant γ is given. In the other three algorithms, some additional conditions on η(r) are supposed, which allow one to uniquely recover η(r) without knowledge of γ : (i) the knowledge of the corresponding constant c4:=η(1)1, (ii) η(1)=1 and the knowledge of the corresponding constant c:=η(1)0, and (iii) η(1)=1, η(1)=0, and the knowledge of the corresponding constant c′′:=η′′(1)0.

Keywords: Transmission eigenvalue; inverse spectral problem; refractive index; reconstruction algorithm; Cauchy data

MSC 2010: 34A55; 34L25; 35R30; 35Q60

References

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About the article

Received: 2015-08-28

Revised: 2015-10-07

Accepted: 2015-10-12

Published Online: 2015-11-28

Published in Print: 2017-02-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11171152

Funding Source: Natural Science Foundation of Jiangsu Province

Award identifier / Grant number: BK 20141392

This work was supported in part by the National Natural Science Foundation of China (11171152) and by the Natural Science Foundation of the Jiangsu Province of China (BK 20141392).


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 1, Pages 23–33, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2015-0081.

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