International Journal of Nonlinear Analysis and Applications

Incomplete inverse problems for the Sturm-Liouville type differential equation with the spectral boundary condition

Document Type : Research Paper

Authors

1 Department of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, 578 Sari, Iran

2 Department of Mathematics, Faculty of Engineering Science, Quchan University of Technology, Quchan, Iran

Abstract

In this study, we examine the inverse problem for the differential equation of the Sturm-Liouville type with the spectral boundary condition in the finite interval. Using Lieberman-Hochstadt's method, we show that if $p(x)$ is prescribed on the half interval  $\left(\frac{\pi}{2},\pi\right)$  then a single spectrum suffices to determine $p(x)$ on $(0,\pi)$. Moreover, applying Gesztesy-Simon's method, we demonstrate that if $p(x)$  is assumed over the given segment $[\pi/2(1 - \theta), \pi]$ where $\theta \in (0, 1),$ a finite number of the spectrum is enough to give $p(x)$ on $(0, \pi)$.

Keywords

[1] V.A. Ambartsumyan, U¨ ber eine frage der eigenwerttheorie, Z. Phys. 53 (1929), 690–695.
[2] R.S. Anderssen, The effect of discontinuous in density and shear velocity on the asymptotic overtone structure of torsional eigenfrequences of the earth, Geophys. J. Royal Astronom. Soc. 50 (1997), 303–309.
[3] G. Borg, Eine Umkehrung der Sturm-Liouville schen Eigenwertaufgabe, Acta Math. 78 (1946), 1–96.
[4] Y. Cakmak and S. Isık, Half inverse problem for the impulsive diffusion operator with discontinuous coefficient, Filomat 30 (2016), no. 1, 157–168.
[5] G. Freiling and V.A. Yurko, Inverse Sturm-Liouville Problems and their Applications, NOVA Science Publ., New York, 2001.
[6] F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, Trans. Amer. Math. Soc. 352 (2000), 2765–2787.
[7] O.H. Hald, Discontinuous inverse eigenvalue problems, Commun. Pure Appl. Math. 37 (1984), no. 5, 539–577.
[8] H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978) 676-680.
[9] Y. Khalili and D. Baleanu, Recovering differential pencils with spectral boundary conditions and spectral jump conditions, J. Inequal. and Appl. 2020 (2020), 262, https://doi.org/10.1186/s13660-020-02537-z.
[10] Y. Khalili, N. Kadkhoda, and D. Baleanu, Inverse problems for the impulsive Sturm–Liouville operator with jump conditions, Inve. Probl. Sci. Eng. 27 (2019), no. 10, 1442–1450.
[11] Y. Khalili, M. Yadollahzadeh, and M.K. Moghadam, Half inverse problems for the impulsive operator with eigenvalue-dependent boundary conditions, Electronic J. Differ. Equ. 190 (2017), 1–5.
[12] H. Koyunbakan and E. Panakhov, Half-inverse problem for diffusion operators on the finite interval, J. Math. Anal. Appl. 326 (2007), 1024–1030.
[13] RJ. Kruger, Inverse problems for nonabsorbing media with discontinuous material properties, J. Math. Phys. 23 (1982), no. 3, 396–404.
[14] F.R. Lapwood and T. Usami, Free Oscillation of the Earth, Cambridge University Press, Cambridge, 1981.
[15] A.S. Ozkan and B. Keskin, Inverse nodal problems for Sturm-Liouville equation with eigenparameter-dependent boundary and jump conditions, Inve. Probl. Sci. Eng. 23 (2015), no. 8, 1306–1312.
[16] A.S. Ozkan and B. Keskin, Uniqueness theorems for an impulsive Sturm-Liouville boundary value problem, Appl. Math. J. Chinese Univ. 27 (2012), 428–434.
[17] Y.P. Wang, A uniqueness theorem for Sturm-Liouville operators with eigenparameter dependent boundary conditions, Tamking J. Math. 43 (2012), 145–152.
[18] Y.P. Wang, Inverse problems for Sturm-Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter, Math. Meth. Appl. Sci. 36 (2013), 857–868.
[19] Y.P. Wang and C.T. Shieh, Inverse problems for Sturm-Liouville equations with boundary conditions linearly dependent on the spectral parameter from partial information, Results. Math. 65 (2014), 105–119.
[20] Y.P. Wang, C.T. Shieh, and Y.T. Ma, Inverse spectral problems for Sturm-Liouville operators with partial information, Appl. Math. Lett. 26 (2013), 1175–1181.
[21] Y.P. Wang, C.F. Yang, and Z.Y. Huang, Half inverse problem for Sturm-Liouville operators with boundary conditions dependent on the spectral parameter, Turk. J. Math. 37 (2013), no. 3, 445-454.
[22] C.F. Yang, A uniqueness theorem from partial transmission eigenvalues and potential on a subdomain, Math. Meth. Appl. Sci. 39 (2016), 527–532.
[23] C.F. Yang, Inverse problems for the Sturm-Liouville operator with discontinuity, Inve. Probl. Sci. Eng. 22 (2014), 232–244.
[24] C.F. Yang and A. Zettl, Half inverse problems for quadratic pencils of Sturm-Liouville operators, Taiwanese J. Math. 16 (2012), no. 5 1829–1846.
International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 4
April 2025
Pages 103-108
  • Receive Date: 10 January 2024
  • Accept Date: 12 March 2024