September  2019, 9(3): 509-515. doi: 10.3934/mcrf.2019023

Determining the shape of a solid of revolution

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

Received  June 2017 Revised  May 2018 Published  April 2019

We show how to reconstruct the shape of a solid of revolution by measuring its temperature on the boundary. This inverse problem reduces to finding a coefficient of a parabolic equation from values of the trace of its solution on the boundary. This is achieved by using the inverse spectral theory of the string, as developed by M.G. Krein, which provides uniqueness and also a reconstruction algorithm.

Citation: Amin Boumenir. Determining the shape of a solid of revolution. Mathematical Control and Related Fields, 2019, 9 (3) : 509-515. doi: 10.3934/mcrf.2019023
References:
[1]

S. A. AvdoninV.S. Mikhaylov and K. Ramdani, Reconstructing the potential for the one-dimensional Schrodinger equation from boundary measurements, IMA J. Math. Control Inform., 31 (2014), 137-150.  doi: 10.1093/imamci/dnt009.

[2]

S. A. Avdonin and V.S. Mikhaylov, Spectral estimation problem in infinite dimensional spaces. Zap. Nauchn. Semin. POMI, 422, 5-18, 2014, J. Math. Sci. (N.Y.), 206 (2015), 3,231–240. doi: 10.1007/s10958-015-2307-7.

[3]

A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation, Proc. Amer. Math. Soc., 138 (2010), 11, 3911–3921. doi: 10.1090/S0002-9939-2010-10297-6.

[4]

A. Boumenir and Vu Kim Tuan, Recovery of the heat coefficient by two measurements, Inverse Problems and Imaging, 5 (2011), 4,775–791. doi: 10.3934/ipi.2011.5.775.

[5]

A. Boumenir and Vu Kim Tuan, Recovery of the heat equation from a single boundary measurement, Applicable Analysis, 10 (2018), 1667-1676.  doi: 10.1080/00036811.2017.1332760.

[6]

S. J. CoxM. Embree and J.M. Hokanson, One can hear the composition of a string: experiments with an inverse eigenvalue problem, SIAM Rev., 54 (2012), 157-178.  doi: 10.1137/080731037.

[7]

H. Dym, and H.P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Dover, 2008.

[8]

I. S. Kac, and M.G. Krein, On the spectral functions of the String, Amer. Math. Soc., Transl., (2), 103 (1974), 19–102. doi: 10.1090/trans2/103/02.

[9]

V. A. Marchenko, Some questions in the theory of one-dimensional linear differential operators of the second order, Six Papers in Analysis, Amer. Math. Soc.(2), 101 (1973), 1–104.

[10]

A. I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Chapman Hall/CRC, Pure and Applied Mathematics, (2000).

[11]

G. Turchetti and G. Sagretti, Stieltjes Functions and Approximation Solutions of an Inverse Problem., Springer Lect. Notes Phys., 85 (1978) 123–33.

[12]

L. Yang. J.N. Yu. Z.C. Deng, An inverse problem of identifying the coefficient of parabolic equation, Applied Mathematical Modeling, 32 10, (2008), 1984–1995. doi: 10.1016/j.apm.2007.06.025.

show all references

References:
[1]

S. A. AvdoninV.S. Mikhaylov and K. Ramdani, Reconstructing the potential for the one-dimensional Schrodinger equation from boundary measurements, IMA J. Math. Control Inform., 31 (2014), 137-150.  doi: 10.1093/imamci/dnt009.

[2]

S. A. Avdonin and V.S. Mikhaylov, Spectral estimation problem in infinite dimensional spaces. Zap. Nauchn. Semin. POMI, 422, 5-18, 2014, J. Math. Sci. (N.Y.), 206 (2015), 3,231–240. doi: 10.1007/s10958-015-2307-7.

[3]

A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation, Proc. Amer. Math. Soc., 138 (2010), 11, 3911–3921. doi: 10.1090/S0002-9939-2010-10297-6.

[4]

A. Boumenir and Vu Kim Tuan, Recovery of the heat coefficient by two measurements, Inverse Problems and Imaging, 5 (2011), 4,775–791. doi: 10.3934/ipi.2011.5.775.

[5]

A. Boumenir and Vu Kim Tuan, Recovery of the heat equation from a single boundary measurement, Applicable Analysis, 10 (2018), 1667-1676.  doi: 10.1080/00036811.2017.1332760.

[6]

S. J. CoxM. Embree and J.M. Hokanson, One can hear the composition of a string: experiments with an inverse eigenvalue problem, SIAM Rev., 54 (2012), 157-178.  doi: 10.1137/080731037.

[7]

H. Dym, and H.P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Dover, 2008.

[8]

I. S. Kac, and M.G. Krein, On the spectral functions of the String, Amer. Math. Soc., Transl., (2), 103 (1974), 19–102. doi: 10.1090/trans2/103/02.

[9]

V. A. Marchenko, Some questions in the theory of one-dimensional linear differential operators of the second order, Six Papers in Analysis, Amer. Math. Soc.(2), 101 (1973), 1–104.

[10]

A. I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Chapman Hall/CRC, Pure and Applied Mathematics, (2000).

[11]

G. Turchetti and G. Sagretti, Stieltjes Functions and Approximation Solutions of an Inverse Problem., Springer Lect. Notes Phys., 85 (1978) 123–33.

[12]

L. Yang. J.N. Yu. Z.C. Deng, An inverse problem of identifying the coefficient of parabolic equation, Applied Mathematical Modeling, 32 10, (2008), 1984–1995. doi: 10.1016/j.apm.2007.06.025.

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