Reconstruction of the coefficients of a star graph from observations of its vertices

Amin Boumenir; Vu Kim Tuan

doi:10.3934/ipi.2018054

Article Contents

2018, Volume 12, Issue 6: 1293-1308. Doi: 10.3934/ipi.2018054

This issue Previous Article A variational model with fractional-order regularization term arising in registration of diffusion tensor image Next Article Stability estimates for a magnetic Schrödinger operator with partial data

Reconstruction of the coefficients of a star graph from observations of its vertices

Amin Boumenir and
Vu Kim Tuan

Department of Mathematics, University of West Georgia, GA 30118, USA

Received: December 2016

Revised: March 2018

Published: December 2018

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

Consider a three-edge star graph, made up of unknown Sturm-Liouville operators on each edge. By using the heat propagation through the graph and measuring the heat transfer occurring at its vertices, we show that we can extract enough spectral data to reconstruct the three Sturm-Liouville operators by using the Gelfand-Levitan theory. Furthermore this reconstruction is achieved by a single measurement provided we use a special initial condition.

Keywords:

Mathematics Subject Classification: 35R30, 35R02, 35K51.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	S. Avdonin and J. Bell, Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Probl. Imaging, 9 (2015), 645-659. doi: 10.3934/ipi.2015.9.645.
[2]	S. Avdonin, J. Bell and K. Nurtazina, Determining distributed parameters in a neuronal cable model on a tree graph, Math. Methods Appl. Sci., 40 (2017), 3973-3981.
[3]	S. Avdonin and S. Nicaise, Source identification for the wave equation on graphs, C. R. Math. Acad. Sci. Paris, 352 (2014), 907-912. doi: 10.1016/j.crma.2014.09.008.
[4]	S. Avdonin, F. Gesztesy and K. Makarov, Spectral estimation and inverse initial boundary value problems, Inverse Probl. Imaging, 4 (2010), 1-9. doi: 10.3934/ipi.2010.4.1.
[5]	S. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem: The case of multiple poles, Math. Control Signals Systems, 22 (2011), 245-265. doi: 10.1007/s00498-010-0052-5.
[6]	S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging, 2 (2008), 1-21. doi: 10.3934/ipi.2008.2.1.
[7]	S. Avdonin, P. Kurasov and M. Nowaczyk, Inverse problems for quantum trees Ⅱ: Recovering matching conditions for star graphs, Inverse Probl. Imaging, 4 (2010), 579-598. doi: 10.3934/ipi.2010.4.579.
[8]	A. Boumenir, Higher approximation of eigenvalues by sampling, BIT, 40 (2000), 215-225. doi: 10.1023/A:1022334806027.
[9]	A. Boumenir, Irregular sampling and the inverse spectral problem, J. Fourier Anal. Appl., 5 (1999), 373-383. doi: 10.1007/BF01259378.
[10]	A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation, Proc. Amer. Math. Soc., 138 (2010), 3911-3921. doi: 10.1090/S0002-9939-2010-10297-6.
[11]	A. Boumenir and V. Kim Tuan, Recovery of the heat coefficient by two measurements, Inverse Problems and Imaging, 5 (2011), 775-791. doi: 10.3934/ipi.2011.5.775.
[12]	R. Carlson and V. Pivovarchik, Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A: Math. Theor., 41 (2008), 145202, 16 pp. doi: 10.1088/1751-8113/41/14/145202.
[13]	I. Kac and V. Pivovarchik, On multiplicity of a quantum graph spectrum, J. Phys. A: Math. Theor., 44 (2011), 105301, 14 pp. doi: 10.1088/1751-8113/44/10/105301.
[14]	H. P. Kramer, A generalized sampling theorem, J. Math. Phys., 38 (1959), 68-72. doi: 10.1002/sapm195938168.
[15]	B. M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.
[16]	B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two of its spectra, Russ. Math. Surveys, 19 (1964), 3-63.
[17]	V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986. doi: 10.1007/978-3-0348-5485-6.
[18]	V. Pivovarchik and H. Woracek, Eigenvalue asymptotics for a star-graph damped vibrations problem, Asymptot. Anal., 73 (2011), 169-185.
[19]	V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32 (2000), 801-819. doi: 10.1137/S0036141000368247.
[20]	V. Kim Tuan and N. Thanh Hong, Interpolation in the Hardy space, Integral Transforms Spec. Funct., 24 (2013), 664-671. doi: 10.1080/10652469.2012.749874.
[21]	A. Zayed, Advances in Shannon's Sampling Theory, CRC Press, 1993.