Abstract
We consider uniqueness of the solution of the inverse problem of determining the coefficient of the one-dimensional wave equation on the real halfline. Necessary conditions of existence of a unique solution of this inverse problem are obtained. A Tikhonov regularizing algorithm is constructed for approximate solution of the inverse problem. The algorithm has an efficient numerical implementation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature Cited
A. S. Alekseev, "Some inverse problems of wave propagation theory," Izv. Akad. Nauk SSSR, Geofiz.,11, 1514–1531 (1962).
A. S. Alekseev, "Inverse dynamic problems of seismology," in: Some Methods and Algorithms for Interpretation of Geophysical Data [in Russian], Nauka, Moscow (1967), pp. 9–84.
A. S. Alekseev and A. G. Megrabov, "The direct and inverse problems for the scattering of plane waves on inhomogeneous transition layers," in: Mathematical Problems of Geophysics [in Russian], No. 3, Izd. VTs Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1972), pp. 8–36.
A. S. Alekseev and A. G. Megrabov, "Inverse problems for a string with the sloping derivative condition on one end and inverse problems of scattering of plane waves on inhomogeneous media," Dokl. Akad. Nauk SSSR,219, No. 2, 308–310 (1974).
I. M. Gel'fand and B. M. Levitan, "On the determination of a differential equation from its spectral function," Izv. Akad. Nauk SSSR, Mat.,15, 309–360 (1951).
A. S. Alekseev, V. I. Dobrinskii, Yu. P. Neprochnov, and G. A. Semenov, Dokl. Akad. Nauk SSSR,228, No. 5, 1053–1056 (1976).
A. Bamberger, D. Chavan, and B. Lelly, "Solution of the inverse problem of seismology using optimal control theory," in: Computational Methods in Applied Mathematics [in Russian], Nauka, Novosibirsk (1982), pp. 108–118.
A. N. Tikhonov, "On regularization of ill-posed problems," Dokl. Akad. Nauk SSSR,153, No. 1, 49–52 (1963).
A. N. Tikhonov and V. Ya. Arsenin, Methods of Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1974).
A. M. Denisov, "On approximate solution of Volterra equation of the first kind," Zh. Vychisl. Mat. Mat. Fiz.,15, No. 4, 1053–1056 (1975).
N. A. Magnitskii, "On approximate solution of functional Volterra equations of the first kind," Vestn. Mosk. Gos. Univ., Ser. Vychisl. Mat. Kibern., No. 4, 72–78 (1977).
E. V. Nikol'skii, "Reflection of plane elastic waves from an arbitrary inhomogeneous layer in case of normal incidence," Zh. Vychisl. Mat. Mat. Fiz., No. 4, 66–078 (1964).
S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press (1973).
H. Gajewski, K. Groeger, and K. Zacharias, Nonlinear Operator Equations [Russian translation], Mir, Moscow (1978).
M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, et al., Approximate Solution of Operator Equations [in Russian], Nauka, Moscow (1969).
Additional information
Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 55–66, 1985.
Rights and permissions
About this article
Cite this article
Baev, A.V. A regularizing algorithm for the solution of the inverse problem for an inhomogeneous string. Comput Math Model 1, 30–38 (1990). https://doi.org/10.1007/BF01128308
Issue Date:
DOI: https://doi.org/10.1007/BF01128308