Yıl: 2021 Cilt: 45 Sayı: 4 Sayfa Aralığı: 1847 - 1870 Metin Dili: İngilizce DOI: 10.3906/mat-2104-40 İndeks Tarihi: 30-06-2022

Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient

Öz:
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.
Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
  • [1] 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; 2013: 1-10.
  • [3] 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; 78: 1-96 (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; 29: 207-224.
  • [13] Keldysh MV. On the eigenvalues and eigenfunctions of some classes of nonselffadjoint equations. Doklady Akademy. Nauk Armian SSSR 1951; 77: 11-14.
  • [14] 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; 378: 549-554.
  • [16] 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: Cambridge University Press, 1981.
  • [18] Levin BYA. Lectures on Entire Functions. Translations of Mathematical Monographs Providence, RI, USA: American Mathematical Society, 1996.
  • [19] Litvinenko ON, Soshnikov VI. The Theory of Heterogeneous Lines and Their Applications in Radio Engineering. Moscow, Russia: Radio, 1964.
  • [20] Marchenko VA. Sturm–Liouville Operators and Their Applications. Kiev, Russia: Naukova Dumka, 1977.
  • [21] Martinyuk O, Pivovarchik V. On the Hochstadt-Lieberman theorem. Inverse Problems 2010; 26 (3).
  • [22] McLaughlin JR. Analytical methods for recovering coefficients in differential equations from spectral data. Society for Industrial and Applied Mathematics Review 1986; 28 (1): 53-72.
  • [23] Meschonav VP, Feldstein AI. Automatic Design of Directional Couplers. Moscow, Russian: Sviaz, 1980.
  • [24] Nabiev AA, Amirov RKh. Integral representations for the solutions of the generalized Schroedinger equation in a finite interval. Advances in Pure Mathematics 2015; 5 (13): 777-795.
  • 25] Rundell W, Sacks PE. Reconstruction techniques for classical inverse Sturm–Liouville problems. Mathematics of Computation 1992; 58 (197): 161-183.
  • [26] Rundell W, Sacks PE. Reconstruction of a radially symmetric potential from two spectral sequences. Journal of Mathematical Analysis and Applications 2001; 264 (2): 354-381.
  • [27] Sakhnovich L. Half-inverse problems on the finite interval. Inverse Problems 2001; 17 (3): 527-532.
  • [28] Shepelsky DG. The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions. Advances in Soviet Mathematics 1997; 19: 303-309.
  • [29] Willis C. Inverse Sturm–Liouville problems with two discontinuties. Inverse Problems 1985; 1 (3): 263-289.
  • [30] Xu XC, Yang CF. Reconstruction of the Sturm–Liouville operator with discontinuities from a particular set of eigenvalues. Applied Mathematics A Journal of Chinese Universities Series B 2018; 33 (2): 225-233.
  • [31] Yamamoto M. Inverse eigenvalue problem for a vibration of a string with viscous drag. Journal of Mathematical Analysis and Applications 1990; 152: 20-34.
  • [32] Yang CF, Yang XP. An interior inverse problem for the Sturm–Liouville operator with discontinuous conditions. Applied Mathematics Letters 2009; 22 (9): 1315-1319.
  • [33] Yang CF. Hochstadt-Lieberman theorem for Dirac operator with eigenparameter dependent boundary conditions. Nonlinear Analysis 2011; 74 (7): 2475-2484.
  • [34] Yang CF, GuoYX. Determination of a differential pencil from interior spectral data. Journal. Mathematical. Analysis Applications 2011; 375: 284-293.
  • [35] Yang CF, Zettl A. Half inverse problems for quadratic pencils of Stur-Liouville operators. Taiwanese Journal Of Mathematics 2012; 16 (5): 1829-1846.
  • [36] Yurko VA. Integral transforms connected with discontinuous boundary value problems. Integral Transforms and Special Functions 2000; 10 (2): 141-164.
  • [37] Zhang R, Xu XC, Yang CF, Bondarenko NP. Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues. Journal of Inverse and Ill-Posed Problems 2019.
APA amirov r, durak s (2021). Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. , 1847 - 1870. 10.3906/mat-2104-40
Chicago amirov rauf,durak sevim Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. (2021): 1847 - 1870. 10.3906/mat-2104-40
MLA amirov rauf,durak sevim Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. , 2021, ss.1847 - 1870. 10.3906/mat-2104-40
AMA amirov r,durak s Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. . 2021; 1847 - 1870. 10.3906/mat-2104-40
Vancouver amirov r,durak s Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. . 2021; 1847 - 1870. 10.3906/mat-2104-40
IEEE amirov r,durak s "Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient." , ss.1847 - 1870, 2021. 10.3906/mat-2104-40
ISNAD amirov, rauf - durak, sevim. "Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient". (2021), 1847-1870. https://doi.org/10.3906/mat-2104-40
APA amirov r, durak s (2021). Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. Turkish Journal of Mathematics, 45(4), 1847 - 1870. 10.3906/mat-2104-40
Chicago amirov rauf,durak sevim Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. Turkish Journal of Mathematics 45, no.4 (2021): 1847 - 1870. 10.3906/mat-2104-40
MLA amirov rauf,durak sevim Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. Turkish Journal of Mathematics, vol.45, no.4, 2021, ss.1847 - 1870. 10.3906/mat-2104-40
AMA amirov r,durak s Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. Turkish Journal of Mathematics. 2021; 45(4): 1847 - 1870. 10.3906/mat-2104-40
Vancouver amirov r,durak s Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient. Turkish Journal of Mathematics. 2021; 45(4): 1847 - 1870. 10.3906/mat-2104-40
IEEE amirov r,durak s "Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient." Turkish Journal of Mathematics, 45, ss.1847 - 1870, 2021. 10.3906/mat-2104-40
ISNAD amirov, rauf - durak, sevim. "Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient". Turkish Journal of Mathematics 45/4 (2021), 1847-1870. https://doi.org/10.3906/mat-2104-40