Abstract
In this paper, we study the inverse coefficient problem of identifying both the mass density
Acknowledgements
I am grateful to Professor Alemdar Hasanov for introducing me to this problem during his research visit at the University of Malta in January 2022, for his encouragement to work on it, and for his valuable scientific feedback and continued support.
References
[1] V. Barcilon, On the solution of inverse eigenvalue problems of high orders, Geophys. J. Int. 39 (1974), 143–154. 10.1111/j.1365-246X.1974.tb05444.xSearch in Google Scholar
[2] V. Barcilon, Inverse problem for the vibrating beam in the free-clamped configuration, Phil. Trans. R. Soc. A 304 (1982), 211–251. Search in Google Scholar
[3] V. Barcilon, Inverse eigenvalue problems, Inverse Problems (Montecatini Terme 1986), Lecture Notes in Math. 1225, Springer, Berlin (1986), 1–51. 10.1007/BFb0072659Search in Google Scholar
[4] O. Baysal and A. Hasanov, Solvability of the clamped Euler–Bernoulli beam equation, Appl. Math. Lett. 93 (2019), 85–90. 10.1016/j.aml.2019.02.006Search in Google Scholar
[5] J.-D. Chang and B.-Z. Guo, Identification of variable spacial coefficients for a beam equation from boundary measurements, Automatica J. IFAC 43 (2007), no. 4, 732–737. 10.1016/j.automatica.2006.11.002Search in Google Scholar
[6] J.-D. Chang and B.-Z. Guo, Application of Ingham–Beurling-type theorems to coefficient identifiability of vibrating systems: Finite time identifiability, Differential Integral Equations 21 (2008), no. 11–12, 1037–1054. 10.57262/die/1355502293Search in Google Scholar
[7] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 2002. Search in Google Scholar
[8] I. M. Gelfand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309–360; translation in Amer. Math. Soc. Transl. Ser. 2 (1951), 253-304. 10.1007/978-3-642-61705-8_24Search in Google Scholar
[9] G. M. L. Gladwell, The inverse problem for the Euler–Bernoulli beam, Proc. Roy. Soc. London Ser. A 407 (1986), no. 1832, 199–218. 10.1098/rspa.1986.0093Search in Google Scholar
[10] A. Hasanov and O. Baysal, Identification of unknown temporal and spatial load distributions in a vibrating Euler–Bernoulli beam from Dirichlet boundary measured data, Automatica J. IFAC 71 (2016), 106–117. 10.1016/j.automatica.2016.04.034Search in Google Scholar
[11] A. Hasanov and O. Baysal, Identification of a temporal load in a cantilever beam from measured boundary bending moment, Inverse Problems 35 (2019), no. 10, Article ID 105005. 10.1088/1361-6420/ab2aa9Search in Google Scholar
[12] A. Hasanov, O. Baysal and C. Sebu, Identification of an unknown shear force in the Euler–Bernoulli cantilever beam from measured boundary deflection, Inverse Problems 35 (2019), no. 11, Article ID 115008. 10.1088/1361-6420/ab2a34Search in Google Scholar
[13] A. Hasanov and H. Itou, A priori estimates for the general dynamic Euler–Bernoulli beam equation: Supported and cantilever beams, Appl. Math. Lett. 87 (2019), 141–146. 10.1016/j.aml.2018.07.038Search in Google Scholar
[14] A. Hasanov and A. Kawano, Identification of unknown spatial load distributions in a vibrating Euler–Bernoulli beam from limited measured data, Inverse Problems 32 (2016), no. 5, Article ID 055004. 10.1088/0266-5611/32/5/055004Search in Google Scholar
[15] A. Hasanov, V. Romanov and O. Baysal, Unique recovery of unknown spatial load in damped Euler–Bernoulli beam equation from final time measured output, Inverse Problems 37 (2021), no. 7, Article ID 075005. 10.1088/1361-6420/ac01fbSearch in Google Scholar
[16] A. Hasanov Hasanoǧlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, 2nd ed., Springer, Cham, 2021. 10.1007/978-3-030-79427-9Search in Google Scholar
[17] C.-H. Huang and C.-C. Shih, An inverse problem in estimating simultaneously the time-dependent applied force and moment of an Euler–Bernoulli beam, CMES Comput. Model. Eng. Sci. 21 (2007), no. 3, 239–254. Search in Google Scholar
[18] V. K. Ivanov, On ill-posed problems, Mat. Sb. (N. S.) 61(103) (1963), 211–223. Search in Google Scholar
[19] S. V. Kalinin and A. Gruverman, Scanning Probe Microscopy. Electrical and Electromechanical Phenomena at the Nanoscale. Vol 1, Springer, New York, 2007. 10.1007/978-0-387-28668-6Search in Google Scholar
[20] A. Kawano, Uniqueness in the determination of unknown coefficients of an Euler–Bernoulli beam equation with observation in an arbitrary small interval of time, J. Math. Anal. Appl. 452 (2017), no. 1, 351–360. 10.1016/j.jmaa.2017.03.019Search in Google Scholar
[21] A. Kawano and A. Morassi, Uniqueness in the determination of loads in multi-span beams and plates, European J. Appl. Math. 30 (2019), no. 1, 176–195. 10.1017/S0956792517000419Search in Google Scholar
[22] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed., Butterworth-Heinemann, New York, 1986. Search in Google Scholar
[23] C.-S. Liu, A Lie-group adaptive differential quadrature method to identify an unknown force in an Euler–Bernoulli beam equation, Acta Mech. 223 (2012), no. 10, 2207–2223. 10.1007/s00707-012-0707-zSearch in Google Scholar
[24] J. R. Mclaughlin, On constructing solutions to an inverse Euler–Bernoulli problem, Inverse Problems of Acoustic and Elastic Waves, Society for Industrial and Applied Mathematics, Philadelphia (1984), 341–347. Search in Google Scholar
[25] S. S. Rao, Vibration of Continuous Systems, John Wiley & Sons, New York, 2007. Search in Google Scholar
[26] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Appl. Math. Sci. 109, Springer, New York, 1995. Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston