We prove that two potential functions on the interval [0, π] , and the parameters in the boundary and impulsive conditions can be determined from a sequence of eigenvalues for two cases: (i) The potentials are given on ( 0, π 4 (1 + α) ) , (ii) The potentials are given on (π 4 (1 + α) , π ) , where 0 < α < 1, respectively.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In this paper, we study the inverse spectral problem for the quadratic differential pencils with discontinuity coefficient on [0, π] with separable boundary conditions and the impulsive conditions at the point x = π 2 .
📊
Key Findings & Data:In this paper, we study the inverse spectral problem for the quadratic differential pencils with discontinuity coefficient on [0, π] with separable boundary conditions and the impulsive conditions at the point x = π 2 .
💡
Conclusion & Impact:We prove that two potential functions on the interval [0, π] , and the parameters in the boundary and impulsive conditions can be determined from a sequence of eigenvalues for two cases: (i) The potentials are given on ( 0, π 4 (1 + α) ) , (ii) The potentials are given on (π 4 (1 + α) , π ) , where 0 < α < 1, respectively.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
In this paper, we study the inverse spectral problem for the quadratic differential pencils with discontinuity coefficient on [0, π] with separable boundary conditions and the impulsive conditions at the point x = π 2 . We prove that two potential functions on the interval [0, π] , and the parameters in the boundary and impulsive conditions can be determined from a sequence of eigenvalues for two cases: (i) The potentials are given on ( 0, π 4 (1 + α) ) , (ii) The potentials are given on (π 4 (1 + α) , π ) , where 0 < α < 1, respectively.
🔖
Cite This Article
AMİROV, R., DURAK, S. (2022). Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. Turkish Journal of Mathematics, Cilt: 45 - Sayı: 4, 1847–1870. https://ajindex.com/articles/607320
Amirov RKh. On Sturm–Liouville operators with discontinuity conditions inside an interval. Journal of Mathematical Analysis and Applications 2006
317 (1): 163-176. 2] Amirov RKh, Nabiev AA. Inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse. Abstract and Applied Analysis 2013
Amirov RKh. On construction of a quadratic Sturm–Liouville operator pencil with impulse from spectral data. Eastern Anatolian Journal of Science 2020
6 (2): 1-8. [4] Bellman R, Cooke KL. Differential-Difference Equations. New York, NY, USA: Academic Press, 1963. [5] Borg G. Eine umkehrung der Sturm–Liouvilleschen eigenwertaufgable. Acta Mathamatica 1946
96 (in German). [6] Buterin SA. On half inverse problem for differential pencils with the spectral parameter in boundary conditions. Tamkang Journal of Mathematics 2011
42 (3): 355-364. [7] Freiling G, Yurko VA. Inverse Sturm–Liouville Problems and Their Applications. New York, NY, USA: Nova Science, 2001. [8] Freiling G, Yurko VA. Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point. Inverse Problems 2002
18 (3): 757-773. [9] Hald OH. Discontinuous inverse eigenvalue problems. Communications on Pure and Applied Mathematics 1984
37 (5): 539-577. [10] Hochstadt H, Lieberman B. An inverse Sturm–Liouville problem with mixed given data. Society for Industrial and Applied Mathematics 1978
34 (4): 676-680. [11] Hryniv RO, Mykytyuk YV. Half-inverse spectral problems for Sturm–Liouville operators with singular potentials. Inverse Problems 2004
20 (5): 1423-1444. [12] Jonas P. On the spectral theory of operators associated with perturbed Klein-Gordon and wave type equations. Journal of Operator Theory 1993
Keldysh MV. On the eigenvalues and eigenfunctions of some classes of nonselffadjoint equations. Doklady Akademy. Nauk Armian SSSR 1951
Kostyuchenko AG, Shkalikov AA. Selfadjoint quadratic operator pencils and elliptic problems. Functional Analysis and its Applications 1983
17 (2): 38-61. [15] Koyunbakan H. Inverse problem for a quadratic pencil of Sturm–Liouville operator. Journal of Mathematical Analysis and Applications 2011
Krueger RJ. Inverse problems for nonabsorbing media with discontinuous material properties. Journal of Mathematical Physics 1982
23 (3): 396-404. [17] Lapwood FR, Usami T. Free Oscillation of The Earth. Cambridge, UK: Cambridg