References
V. G. Romanov, Inverse Problems of Mathematical Physics [in Russian], Nauka, Moscow (1984).
S. I. Kabanikhin, Projection-Difference Methods for Determining Coefficients of Hyperbolic Equations [in Russian], Nauka, Novosibirsk (1988).
I. M. Gelfand and B. M. Levitan, “On determining a differential equation from its spectral function,” Izv. Akad. Nauk SSSR Ser. Mat., 15, No. 4, 309–360 (1951).
M. G. Kreîn, “On a method for effective solution of an inverse boundary value problem,” Dokl. Akad. Nauk SSSR, 94, No. 6, 767–770 (1954).
A. S. Blagoveshchenskiî, “On a local method for solving the nonstationary inverse problem of an inhomogeneous string,” Trudy Mat. Inst. Steklov., 115, 28–38 (1971).
Yu. M. Berezanskiî, “To a uniqueness theorem concerning the inverse problem of spectral analysis for the Schrödinger equation,” Trudy Moskov. Mat. Obshch., 7, 3–51 (1958).
A. A. Androshchuk, “On uniqueness of recovering the two-dimensional Schrödinger equation,” Dokl. Akad. Nauk SSSR, 291, No. 5, 1033–1036 (1986).
M. I. Belishev, “On a certain approach to multidimensional inverse problems for the wave equation,” Dokl. Akad. Nauk SSSR, 297, No. 3, 524–527 (1987).
M. I. Belishev and A. S. Blagoveshchenskiî, “Multidimensional analogs of the Gelfand-Levitan-Kreîn-type equations in an inverse problem for the wave equation,” in: Well-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Novosibirsk, 1992, pp. 50–63.
S. I. Kabanikhin, “On linear regularization of multidimensional inverse problems for hyperbolic equations,” Dokl. Akad. Nauk SSSR, 309, No. 4, 791–795 (1989).
S. I. Kabanikhin and G. B. Bakanov, “Discrete analog of the Gelfand-Levitan equation,” J. Inverse Ill-Posed Probl., 4, No. 5, 409–435 (1996).
Additional information
The research was financially supported by the Russian Foundation for Basic Research (Grant 96-01-01887).
Novosibirsk. Translated from Sibirskiî Matematicheskiî Zhurnal, Vol. 40, No. 2, pp. 307–324, March–April, 1999.
Rights and permissions
About this article
Cite this article
Kabanikhin, S.I., Bakanov, G.B. A discrete analog of the Gelfand-Levitan method in a two-dimensional inverse problem for a hyperbolic equation. Sib Math J 40, 262–280 (1999). https://doi.org/10.1007/s11202-999-0007-6
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11202-999-0007-6