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Reconstruction of Potential in Discrete Sturm–Liouville Problem

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Abstract

In this paper, we consider the Sturm–Liouville problem with Dirichlet conditions in the case of time scales consists isolated points. Then, we obtain discrete Sturm–Liouville problem on a finite interval. We solve the inverse nodal problem, especially give a reconstruction formula for the potential function q.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston (2001)

    Book  Google Scholar 

  2. Hilger, S.: Ein Masskettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD, Universitat Wurzburg, Wurzburg, Germany (1988) (in German)

  3. Kelley, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications, 2nd edn. Academic Press, San Diego (2001)

    MATH  Google Scholar 

  4. Agarwal, R.P., Bohner, M., Wong, P.J.Y.: Eigenvalues and eigenfunctions of discrete conjugate boundary value problems. Comput. Math. Appl. 38(3–4), 159–183 (1999)

    Article  MathSciNet  Google Scholar 

  5. Allahverdiev, B., Eryilmaz, A., Tuna, H.: Dissipative Sturm-Liouville operators with a spectral parameter in the boundary condition on bounded time scales, Electronic. J. Differ. Equ. 95, 1–13 (2017)

    MATH  Google Scholar 

  6. Bala, B., Kablan, A., Manafov, M.: Direct and inverse spectral problems for discrete Sturm-Liouville problem with generalized function potential. Adv. Differ. Equ. 2016, 172 (2016). https://doi.org/10.1186/s13662-016-0898-z

    Article  MathSciNet  MATH  Google Scholar 

  7. Bohner, M.: On disconjugacy for Sturm-Liouville difference equations. J. Differ. Equ. Appl. 2(2), 227–237 (1996)

    Article  MathSciNet  Google Scholar 

  8. Bohner, M.: Asymptotic behavior of discretized Sturm-Liouville eigenvalue problems. J. Differ. Equ. Appl. 3, 289–295 (1998)

    Article  MathSciNet  Google Scholar 

  9. Bohner, M.: Discrete Sturmian theory. Math. Inequ. Appl. 1(3), 375–383 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Bohner, M., Koyunbakan, H.: Inverse problems for Sturm-Liouville difference equations. Filomat 30(5), 1297–1304 (2016)

    Article  MathSciNet  Google Scholar 

  11. Ahlbrandt, C., Bohner, M., Voepel, T.: Variable change for Sturm-Liouville differential expressions on time scales. J. Differ. Equ. Appl. 9(1), 93–107 (2003) (in honour of Professor Allan Peterson on the occasion of his 60th birthday, part II)

  12. Currie, S., Love, A.: Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions II. Adv. Pure Math. 6(10), 625–632 (2016)

    Article  Google Scholar 

  13. Gao, C., Ma, R.: Eigenvalues of discrete Sturm-Liouville problems with eigenparameter dependent boundary conditions. Linear Algebra Appl. 503(15), 100–119 (2016)

    Article  MathSciNet  Google Scholar 

  14. Yilmaz, E., Gulsen, T., Koyunbakan, H.: Conformable fractional Sturm-Liouville equation and some existence results on time scales. Turkish J. Math. 42, 1348–1360 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Ambartsumyan, V.A.: Über eine Frage der Eigenwerttheorie. Zeitschrift für Physik 53, 690–695 (1929)

    Article  Google Scholar 

  16. Buterin, S.A., Shieh, C.T.: Incomplete inverse spectral and nodal problems for differential pencils. Results Math. 62(1–2), 167–179 (2012)

    Article  MathSciNet  Google Scholar 

  17. Hald, O., McLaughlin, J.R.: Solution of the inverse nodal problems. Inverse Probl. 5, 307–347 (1989)

    Article  MathSciNet  Google Scholar 

  18. Hu, Y.T., Bondarenko, N.P., Shieh, C.T., Yang, C.F.: Traces and inverse nodal problems for Dirac-type integro-differential operators on a graph. Appl. Math. Comput. 363, 124606 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Zhang, R., Sat, M., Yang, C.F.: Inverse nodal problem for the Sturm-Liouville operator with a weightAppl. Math. J. Chinese Univ. 35(2), 193–202 (2020)

    Article  MathSciNet  Google Scholar 

  20. Hu, Y.T., Bondarenko, N.P., Yang, C.F.: Traces and inverse nodal problem for Sturm-Liouville operators with frozen argument. Appl. Math. Lett. 102, 106096 (2020) (7pp)

  21. Yang, C.F., Xu, X.C., Buterin, S.A.: Solution to the interior transmission problem using nodes on a subinterval as input data. Nonlinear Anal. Real World Appl. 35, 20–29 (2017)

    Article  MathSciNet  Google Scholar 

  22. Law, C.K., Yang, C.F.: Reconstruction of the potential function and its derivatives using nodal data. Inverse Probl. 14, 299–312 (1998)

    Article  Google Scholar 

  23. McLaughlin, J.R.: Inverse spectral theory using nodal points as data-a uniqueness result. J. Differ. Equ. 73, 354–362 (1988)

    Article  MathSciNet  Google Scholar 

  24. Sadovnichii, V.A., Sultanaev, Y.T., Akhtyamov, A.M.: Solvability theorems for an inverse nonself-adjoint Sturm-Liouville problem with nonseparated boundary conditions. Differ. Equ. 51(6), 717–725 (2015)

    Article  MathSciNet  Google Scholar 

  25. Shen, C.L.: On the nodal sets of the eigenfunctions of the string equations. SIAM J. Math. Anal. 19(89), 1419–1424 (1988)

    Article  MathSciNet  Google Scholar 

  26. Kratz, W.: Quadratic Functionals in Variational Analysis and Control Theory, volume 6 of Mathematical Topics. Akademie Verlag (1995)

  27. Paine, J.W., Anderssen, R.S., De Hoog, F.R.: On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems. Computing 26, 123–139 (1981)

    Article  MathSciNet  Google Scholar 

  28. Anderssen, R.S., De Hoog, F.R.: On the correction of finite difference eigenvalue approximation for Sturm-Liouville problems with general boundary conditions. BIT 24, 401–402 (1984)

    Article  MathSciNet  Google Scholar 

  29. Anderssen, R.S., De Hoog, F.R.: Asymtotic formulas for discrete eigenvalue problems in Liouville normal form. Math. Models Methods Appl. Sci. 11(1), 43–56 (2001)

    Article  MathSciNet  Google Scholar 

  30. Elaydi, S.: An Introduction to Difference Equations. Third Edition Springer (1996)

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Correspondence to Hikmet Koyunbakan.

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Koyunbakan, H. Reconstruction of Potential in Discrete Sturm–Liouville Problem. Qual. Theory Dyn. Syst. 21, 13 (2022). https://doi.org/10.1007/s12346-021-00548-9

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