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Discontinuous Density Function Identification

Year 2020, Volume: 12 Issue: 1, 45 - 48, 29.06.2020

Abstract

The work is devoted to the identification step density function of a string. The inverse problem consists of recovering constant densities $ \rho_{i}$ of eigenvalue problem. It is shown that if we use only the natural frequencies of the boundary value problem itself to restore the step density, then this inverse problem has an infinite number of solutions $ \rho = \left( \rho_{1}, \rho_{2}, \dots , \rho_{n} \right) $ in $ {\mathbb{R}}^{n} $ and unique solution in a sufficiently small area $ \Omega \subset \mathbb{R}^{n}$. For the uniqueness of the recovery of the step density of a string, the natural frequencies of one boundary value problem are not enough. We need to use the natural frequencies of the two boundary problems. To uniquely reconstruct a step density function, we need to use natural frequencies of the boundary value problem itself and natural frequencies of another boundary problem, which differs from the first one only by one boundary condition. In M. Krein uniqueness theorems, to restore the continuous density function, we used all the eigenvalues of the two problems. In contrast to the M. Krein uniqueness theorems, for the uniqueness of the recovery of the n-step density function, we need a finite number of eigenvalues.

Supporting Institution

Russian Foundation of Basic Research

Project Number

060100258,0901403

Thanks

The reported research was funded by Russian Foundation for Basic Research, the government of the region of the Republic of Bashkortostan (projects 18-51-06002-Az a 18-01-00250 a, 17-41-020230-r a), and the Science Development Fund under the President of the Republic of Azerbaijan (project on the 1st Azerbaijan-Russian International Grant Competition (EIF-BGM-4-RFTF-1/2017)).

References

  • Akhmedova, E.N., {\it On representation of a solution of Sturm-Liouville equation with discontinuous coefficients}, Proceedings of IMM of NAS of Azerbaijan, \textbf{16(24)}(2002), 5--9.
  • Akhmedova, E.N., H\"{u}seynov H.M {\it On eigenvalues and eigenfunctions of one class of Sturm-Liouville operators with discontinuous coefficients}, Transactions of NAS of Azerbaijan, \textbf{23(4)}(2003), 7--18.
  • Akhmedova, E.N., The definition of one class of Sturm-Liouville operators with discontinuous coefficients by Weyl function, Proceedings of IMM of NAS of Azerbaijan, 2005, \textbf{22(30)}(2005), 3--8.
  • Carlson, R. {\it An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients by Weyl function}, Proceedings of IMM Of NAS of Azerbaijan, 2005, \textbf{22(30)}(2005), 3--8.
  • Gasymov, M.G., The Direct and Inverse Problem of Spectral Analysis for a Class of Equations with a Discontinuous Coefficient, Non-Classical Methods in Geophysics, M. M. Laurent'ev, Ed., Novosibirsk, pp.~37-44, 1977.
  • Kadchenko, S.I., {\it A numerical method for solving inverse problems generated by perturbed self-adjoint operators}, Bulletin of the South Ural State University. Series: Mathematical Modeling and Programming, \textbf{60(4)}(2013), 15--25.
  • Krein, M.G., {\it Determination of the density of an onhomogeneous symmetric cord by its frequency spectrum}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{796}(1951), 345--348.
  • Krein, M.G., {\it On inverse problems for an onhomogeneous cord}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{82}(1952), 669--672.
  • Krein, M.G. {\it On a generalization of investigations of Stiltjes}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{87}(1952), 881--884.
  • Krein, M.G.,{\it On some cases of effective determination of the density of aninhomogeneous cord from its spectral function}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{93}(1953), 617--620.
  • Krein, M.G., {\it On a method of effective solution of an inverse boundary problem, }, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{94}(1954), 987--990.
  • Levitan, B.M., Inverse Sturm-Liouville Problems, Utrecht: VNU Science Press, 1987.
  • Marchenko, V.A., Sturm-Liouville Operators and Applications, Basel, Boston, Stuttgart: Birkhauser, 1986.
  • Mizrak, O., Mamedov, Kh.R., Akhtyamov, A.M., {\it Characteristic properties of scattering data of a boundary value problem}, Filomat, \textbf{31(12)}(2017), 3945--3951.
  • Sadovnichii, V.A., Dubrovskii, V.V., Kadchenko, S.I., Kravchenko, V.F., {\it Computation of lower eigenvalues of the boundary value problem on the hydrodynamic stability of poiseuille flow in a round tube}, Differential Equations, \textbf{34(1)}(1998), 49--53.
  • Sadovnichii, V.A., Sultanaev, Ya.T., Akhtyamov, A.M., {\it Solvability theorems for an inverse nonself-adjoint Sturm--Liouville problem with nonseparated boundary conditions}, Differential Equations, \textbf{51(6)}(2015), 717--725.
  • Sadovnichii, V.A., Sultanaev, Y.T., Akhtyamov, A.M., {\it Well-posedness of the inverse Sturm--Liouville problem with nonseparated boundary conditions}, Doklady Mathematical Sciences, \textbf{69(2)}(2004), 253--256.
  • Shepelsky, D.J., {\it The inverse problem of reconstruction of the Medium's conductivity in a class of increasing functions}, Adv. Sov. Math., \textbf{19}(1994), 209--231.
  • \c{S}en, E., Mukhtarov, O.S, {\it Spectral properties of discontinuous Sturm-Liouville problems with a finite number of transmission conditions}, Mediterr. J. Math., \textbf{13}(2016), 153--170.
  • Yurko, V.A., {\it On the inverse problem for differential operators on a finite interval with complex weights}, Math. Notes, \textbf{105}(2019), 301--306.
  • Yurko, V.A., {\it Inverse problems for arbitrary order differential operators with discontinuties in an interior point}, Results Math., \textbf{73(25)}(2018).
Year 2020, Volume: 12 Issue: 1, 45 - 48, 29.06.2020

Abstract

Project Number

060100258,0901403

References

  • Akhmedova, E.N., {\it On representation of a solution of Sturm-Liouville equation with discontinuous coefficients}, Proceedings of IMM of NAS of Azerbaijan, \textbf{16(24)}(2002), 5--9.
  • Akhmedova, E.N., H\"{u}seynov H.M {\it On eigenvalues and eigenfunctions of one class of Sturm-Liouville operators with discontinuous coefficients}, Transactions of NAS of Azerbaijan, \textbf{23(4)}(2003), 7--18.
  • Akhmedova, E.N., The definition of one class of Sturm-Liouville operators with discontinuous coefficients by Weyl function, Proceedings of IMM of NAS of Azerbaijan, 2005, \textbf{22(30)}(2005), 3--8.
  • Carlson, R. {\it An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients by Weyl function}, Proceedings of IMM Of NAS of Azerbaijan, 2005, \textbf{22(30)}(2005), 3--8.
  • Gasymov, M.G., The Direct and Inverse Problem of Spectral Analysis for a Class of Equations with a Discontinuous Coefficient, Non-Classical Methods in Geophysics, M. M. Laurent'ev, Ed., Novosibirsk, pp.~37-44, 1977.
  • Kadchenko, S.I., {\it A numerical method for solving inverse problems generated by perturbed self-adjoint operators}, Bulletin of the South Ural State University. Series: Mathematical Modeling and Programming, \textbf{60(4)}(2013), 15--25.
  • Krein, M.G., {\it Determination of the density of an onhomogeneous symmetric cord by its frequency spectrum}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{796}(1951), 345--348.
  • Krein, M.G., {\it On inverse problems for an onhomogeneous cord}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{82}(1952), 669--672.
  • Krein, M.G. {\it On a generalization of investigations of Stiltjes}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{87}(1952), 881--884.
  • Krein, M.G.,{\it On some cases of effective determination of the density of aninhomogeneous cord from its spectral function}, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{93}(1953), 617--620.
  • Krein, M.G., {\it On a method of effective solution of an inverse boundary problem, }, Dokl. Akad. Nauk SSSR, (in Russian), \textbf{94}(1954), 987--990.
  • Levitan, B.M., Inverse Sturm-Liouville Problems, Utrecht: VNU Science Press, 1987.
  • Marchenko, V.A., Sturm-Liouville Operators and Applications, Basel, Boston, Stuttgart: Birkhauser, 1986.
  • Mizrak, O., Mamedov, Kh.R., Akhtyamov, A.M., {\it Characteristic properties of scattering data of a boundary value problem}, Filomat, \textbf{31(12)}(2017), 3945--3951.
  • Sadovnichii, V.A., Dubrovskii, V.V., Kadchenko, S.I., Kravchenko, V.F., {\it Computation of lower eigenvalues of the boundary value problem on the hydrodynamic stability of poiseuille flow in a round tube}, Differential Equations, \textbf{34(1)}(1998), 49--53.
  • Sadovnichii, V.A., Sultanaev, Ya.T., Akhtyamov, A.M., {\it Solvability theorems for an inverse nonself-adjoint Sturm--Liouville problem with nonseparated boundary conditions}, Differential Equations, \textbf{51(6)}(2015), 717--725.
  • Sadovnichii, V.A., Sultanaev, Y.T., Akhtyamov, A.M., {\it Well-posedness of the inverse Sturm--Liouville problem with nonseparated boundary conditions}, Doklady Mathematical Sciences, \textbf{69(2)}(2004), 253--256.
  • Shepelsky, D.J., {\it The inverse problem of reconstruction of the Medium's conductivity in a class of increasing functions}, Adv. Sov. Math., \textbf{19}(1994), 209--231.
  • \c{S}en, E., Mukhtarov, O.S, {\it Spectral properties of discontinuous Sturm-Liouville problems with a finite number of transmission conditions}, Mediterr. J. Math., \textbf{13}(2016), 153--170.
  • Yurko, V.A., {\it On the inverse problem for differential operators on a finite interval with complex weights}, Math. Notes, \textbf{105}(2019), 301--306.
  • Yurko, V.A., {\it Inverse problems for arbitrary order differential operators with discontinuties in an interior point}, Results Math., \textbf{73(25)}(2018).
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Volkan Ala

Hanlar Reşidoğlu

Azamat M. Akhtyamov This is me 0000-0002-2080-6648

Project Number 060100258,0901403
Publication Date June 29, 2020
Published in Issue Year 2020 Volume: 12 Issue: 1

Cite

APA Ala, V., Reşidoğlu, H., & Akhtyamov, A. M. (2020). Discontinuous Density Function Identification. Turkish Journal of Mathematics and Computer Science, 12(1), 45-48.
AMA Ala V, Reşidoğlu H, Akhtyamov AM. Discontinuous Density Function Identification. TJMCS. June 2020;12(1):45-48.
Chicago Ala, Volkan, Hanlar Reşidoğlu, and Azamat M. Akhtyamov. “Discontinuous Density Function Identification”. Turkish Journal of Mathematics and Computer Science 12, no. 1 (June 2020): 45-48.
EndNote Ala V, Reşidoğlu H, Akhtyamov AM (June 1, 2020) Discontinuous Density Function Identification. Turkish Journal of Mathematics and Computer Science 12 1 45–48.
IEEE V. Ala, H. Reşidoğlu, and A. M. Akhtyamov, “Discontinuous Density Function Identification”, TJMCS, vol. 12, no. 1, pp. 45–48, 2020.
ISNAD Ala, Volkan et al. “Discontinuous Density Function Identification”. Turkish Journal of Mathematics and Computer Science 12/1 (June 2020), 45-48.
JAMA Ala V, Reşidoğlu H, Akhtyamov AM. Discontinuous Density Function Identification. TJMCS. 2020;12:45–48.
MLA Ala, Volkan et al. “Discontinuous Density Function Identification”. Turkish Journal of Mathematics and Computer Science, vol. 12, no. 1, 2020, pp. 45-48.
Vancouver Ala V, Reşidoğlu H, Akhtyamov AM. Discontinuous Density Function Identification. TJMCS. 2020;12(1):45-8.