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Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map
Recovery of the heat coefficient by two measurements
1. | Department of Mathematics, Kuwait University, Safat -13060, Kuwait |
2. | Department of Mathematics, University of West Georgia, Carrollton, GA 30118, United States |
References:
[1] |
A. L. Andrew, Computing Sturm-Liouville potentials from two spectra,, Inverse Problems, 22 (2006), 2069.
doi: 10.1088/0266-5611/22/6/010. |
[2] |
S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995).
|
[3] |
S. A. Avdonin, M. I. Belishev and Yu. Rozhkov, The BC-method in the inverse problem for the heat equation,, J. Inv. Ill-Posed Probl., 5 (1997), 309.
doi: 10.1515/jiip.1997.5.4.309. |
[4] |
S. A. Avdonin and M. I. Belishev, Boundary control and dynamical inverse problem for nonselfadjoint Sturm-Liouville operator (BC-method),, in, 25 (1996), 429.
|
[5] |
S. A. Avdonin, S. Lenhart and V. Protopopescu, Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method. Inverse problems: Modeling and simulation,, J. Inv. Ill-Posed Probl., 13 (2005), 317.
doi: 10.1515/156939405775201718. |
[6] |
M. I. Belishev, A canonical model of a dynamical system with boundary control in the inverse heat conduction problem, (in Russian),, Algebra i Analiz, 7 (1995), 3.
|
[7] |
A. Boumenir, The recovery of analytic potentials,, Inverse Problems, 15 (1999), 1405.
doi: 10.1088/0266-5611/15/6/302. |
[8] |
A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation,, Proc. Am. Math. Soc., 138 (2010), 3911.
doi: 10.1090/S0002-9939-2010-10297-6. |
[9] |
A. Boumenir and Vu Kim Tuan, Recovery of a heat equation by four measurements at one end,, Numer. Funct. Anal. Optim., 31 (2010), 155.
doi: 10.1080/01630560903574993. |
[10] |
R. H. Fabiano, R. Knobel and B. D. Lowe, A finite difference algorithm for an inverse Sturm-Liouville problem,, IMA J. Appl. Math., 15 (1995), 75.
|
[11] |
V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations,, Arch. Rational Mech. Anal., 124 (1993), 1.
doi: 10.1007/BF00392201. |
[12] |
V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).
|
[13] |
A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001).
|
[14] |
B. M. Levitan, "Inverse Sturm-Liouville Problems,", Translated from the Russian by O. Efimov, (1987).
|
[15] |
B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra,, Uspehi Mat. Nauk, 19 (1964), 3.
|
[16] |
B. M. Levitan and I. S. Sargsjan, "Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators,", Translated from the Russian by Amiel Feinstein, (1975).
|
[17] |
V. A. Marčenko, Some questions in the theory of one-dimensional linear differential operators of the second order (1-104), in, American Mathematical Society Translations, (1973). Google Scholar |
[18] |
J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data,, SIAM Rev., 28 (1986), 53.
doi: 10.1137/1028003. |
[19] |
J. R. McLaughlin, Solving inverse problems with spectral data,, in, (2000), 169.
doi: 10.1007/978-3-7091-6296-5_10. |
[20] |
B. D. Lowe and W. Rundell, The determination of a coefficient in a parabolic equation from input sources,, IMA J. Appl. Math., 52 (1994), 31.
doi: 10.1093/imamat/52.1.31. |
[21] |
B. D. Lowe, M. Pilant and W. Rundell, The recovery of potentials from finite spectral data,, SIAM J. Math. Anal., 23 (1992), 482.
doi: 10.1137/0523023. |
[22] |
Y. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,, IEEE Transactions of Acoustics, 38 (1990), 814.
doi: 10.1109/29.56027. |
[23] |
Y. Hua, A. B. Gershman and Q. Cheng, "High-Resolution and Robust Signal Processing,", Marcel Dekker, (2004). Google Scholar |
[24] |
T. K. Sarkar, M. C. Wicks, M. Salazar-Palma and R. J. Bonneau, "Smart Antennas,", John Wiley & Sons, (2003).
doi: 10.1002/0471722839. |
show all references
References:
[1] |
A. L. Andrew, Computing Sturm-Liouville potentials from two spectra,, Inverse Problems, 22 (2006), 2069.
doi: 10.1088/0266-5611/22/6/010. |
[2] |
S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995).
|
[3] |
S. A. Avdonin, M. I. Belishev and Yu. Rozhkov, The BC-method in the inverse problem for the heat equation,, J. Inv. Ill-Posed Probl., 5 (1997), 309.
doi: 10.1515/jiip.1997.5.4.309. |
[4] |
S. A. Avdonin and M. I. Belishev, Boundary control and dynamical inverse problem for nonselfadjoint Sturm-Liouville operator (BC-method),, in, 25 (1996), 429.
|
[5] |
S. A. Avdonin, S. Lenhart and V. Protopopescu, Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method. Inverse problems: Modeling and simulation,, J. Inv. Ill-Posed Probl., 13 (2005), 317.
doi: 10.1515/156939405775201718. |
[6] |
M. I. Belishev, A canonical model of a dynamical system with boundary control in the inverse heat conduction problem, (in Russian),, Algebra i Analiz, 7 (1995), 3.
|
[7] |
A. Boumenir, The recovery of analytic potentials,, Inverse Problems, 15 (1999), 1405.
doi: 10.1088/0266-5611/15/6/302. |
[8] |
A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation,, Proc. Am. Math. Soc., 138 (2010), 3911.
doi: 10.1090/S0002-9939-2010-10297-6. |
[9] |
A. Boumenir and Vu Kim Tuan, Recovery of a heat equation by four measurements at one end,, Numer. Funct. Anal. Optim., 31 (2010), 155.
doi: 10.1080/01630560903574993. |
[10] |
R. H. Fabiano, R. Knobel and B. D. Lowe, A finite difference algorithm for an inverse Sturm-Liouville problem,, IMA J. Appl. Math., 15 (1995), 75.
|
[11] |
V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations,, Arch. Rational Mech. Anal., 124 (1993), 1.
doi: 10.1007/BF00392201. |
[12] |
V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).
|
[13] |
A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001).
|
[14] |
B. M. Levitan, "Inverse Sturm-Liouville Problems,", Translated from the Russian by O. Efimov, (1987).
|
[15] |
B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra,, Uspehi Mat. Nauk, 19 (1964), 3.
|
[16] |
B. M. Levitan and I. S. Sargsjan, "Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators,", Translated from the Russian by Amiel Feinstein, (1975).
|
[17] |
V. A. Marčenko, Some questions in the theory of one-dimensional linear differential operators of the second order (1-104), in, American Mathematical Society Translations, (1973). Google Scholar |
[18] |
J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data,, SIAM Rev., 28 (1986), 53.
doi: 10.1137/1028003. |
[19] |
J. R. McLaughlin, Solving inverse problems with spectral data,, in, (2000), 169.
doi: 10.1007/978-3-7091-6296-5_10. |
[20] |
B. D. Lowe and W. Rundell, The determination of a coefficient in a parabolic equation from input sources,, IMA J. Appl. Math., 52 (1994), 31.
doi: 10.1093/imamat/52.1.31. |
[21] |
B. D. Lowe, M. Pilant and W. Rundell, The recovery of potentials from finite spectral data,, SIAM J. Math. Anal., 23 (1992), 482.
doi: 10.1137/0523023. |
[22] |
Y. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,, IEEE Transactions of Acoustics, 38 (1990), 814.
doi: 10.1109/29.56027. |
[23] |
Y. Hua, A. B. Gershman and Q. Cheng, "High-Resolution and Robust Signal Processing,", Marcel Dekker, (2004). Google Scholar |
[24] |
T. K. Sarkar, M. C. Wicks, M. Salazar-Palma and R. J. Bonneau, "Smart Antennas,", John Wiley & Sons, (2003).
doi: 10.1002/0471722839. |
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