Solving Symmetric Inverse Sturm–Liouville Problem Using Chebyshev Polynomials

  • A. Neamaty
  • Sh. Akbarpoor
  • E. YilmazEmail author


In this study, we consider Sturm–Liouville equation having a symmetric potential function under the separated boundary conditions on a finite interval. Then, we use Chebyshev polynomials of the first kind for calculating the approximate solution of the inverse Sturm–Liouville problem. Finally, we present the numerical results by providing some examples.


Inverse problem Sturm–Liouville operator symmetric potential Chebyshev polynomials 

Mathematics Subject Classification

34K10 34K29 34K28 



  1. 1.
    Aceto, L., Ghelardoni, P., Magherini, C.: Boundary value methods for the reconstruction of Sturm–Liouville potentials. Appl. Math. Comput. 219, 2960–2974 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambartsumyan, V.A.: Über eine frage der eigenwerttheorie. Zeitschrift für Physik. 53, 690–695 (1929)CrossRefGoogle Scholar
  3. 3.
    Andrew, A.L.: Numerov’s method for inverse Sturm–Liouville problem. Inverse Probl. 21, 223–238 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Andrew, A.L.: Computing Sturm–Liouville potentials from two spectra. Inverse Probl. 22, 2069–2081 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Böckmann, C., Kammanee, A.: Broyden method for inverse non-symmetric Sturm–Liouville problems. BIT 51, 513–528 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Browne, P.J., Sleeman, B.D.: A uniqueness theorem for inverse eigenparameter dependent Sturm–Liouville problems. Inverse Probl. 13, 1453–1462 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Drignei, M.C.: A Newton-type method for solving an inverse Sturm–Liouville problem. Inverse Probl. Sci. Eng. 23(5), 851–883 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Efremova, L., Freiling, G.: Numerical solution of inverse spectral problems for Sturm–Liouville operators with discontinuous potentials. Cent. Eur. J. Math. 11, 2044–2051 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fabiano, R.H., Knobel, R., Lowe, B.D.: A finite-difference algorithm for an inverse Sturm–Liouville problem. IMA J. Numer. Anal. 15, 75–88 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Freiling, G., Yurko, V.: Inverse Sturm–Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001)zbMATHGoogle Scholar
  11. 11.
    Gao, Q., Cheng, X., Huang, Z.: Modified Numerov’s method for inverse Sturm–Liouville problems. J. Comput. Appl. Math. 253, 181–199 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gladwell, G.M.L.: The application of Schur’s algorithm to an inverse eigenvalue problem. Inverse Probl. 7, 557–565 (1991)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hald, O.H.: The inverse Sturm–Liouville problem and the Rayleigh–Ritz method. Math. Comput. 32, 687–705 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ignatiev, M., Yurko, V.: Numerical methods for solving inverse Sturm–Liouville problems. Result Math. 52, 63–74 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kammanee, A., Böckmann, C.: Boundary value method for Sturm–Liouville problems. Appl. Math. Comput. 214, 342–352 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kobayashi, M.: A uniqueness proof for discontinuous inverse Sturm–Liouville problems with symmetric potentials. Inverse Probl. 5, 767–781 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Levitan, B.M., Sargsjan, I.S.: Introduction to Spectral Theory: Self Adjoint Ordinary Differential Operators. American Mathematical Society, Providence, RI (1975)CrossRefGoogle Scholar
  18. 18.
    Lowe, B.D., Pilant, M., Rundell, W.: The recovery of potentials from finite spectral data. SIAM J. Math. Anal. 23, 482–504 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Marchenko, V.A., Maslov, K.V.: Stability of the problem of recovering the Sturm–Liouville operator from the spectral function. Math. USSR Sbornik. 81, 475–502 (1970)CrossRefGoogle Scholar
  20. 20.
    Neamaty, A., Khalili, Y.: Determination of a differential operator with discontinuity from interior spectral data. Inverse Probl. Sci. Eng. 22, 1002–1008 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pivovarchik, V.: Direct and inverse three-point Sturm–Liouville problems with parameter-dependent boundary conditions. Asymptot. Anal. 26, 219–238 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Volume 130 of Pure and Applied Mathematics. Academic Press, Inc, Boston, MA (1987)zbMATHGoogle Scholar
  23. 23.
    Rafler, M., Böckmann, C.: Reconstruction method for inverse Sturm–Liouville problems with discontinuous potentials. Inverse Probl. 23, 933–946 (2007)CrossRefGoogle Scholar
  24. 24.
    Rashed, M.T.: Numerical solution of a special type of integro-differential equations. Appl. Math. Comput. 143, 73–88 (2003)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Röhrl, N.: A least-squares functional for solving inverse Sturm–Liouville problems. Inverse Probl. 21, 2009–2017 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Röhrl, N.: Recovering boundary conditions in inverse Sturm-Liouville problems. In: Recent Advances in Differential Equations and Mathematical Physics. Vol. 412, Contemporary mathematics. Providence (RI): Amer. Math. Soc; 2006. p. 263–270. arXiv: math.NA/0601031
  27. 27.
    Rundell, W., Sacks, P.E.: Reconstruction techniques for classical inverse Sturm–Liouville problems. Math. Comput. 58, 161–183 (1992)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sacks, P.E.: An iterative method for the inverse Dirichlet problem. Inverse Probl. 4, 1055–1069 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shahriari, M., Jodayree, A., Teschl, G.: Uniqueness for inverse Sturm–Liouville problems with a finite number of transmission conditions. J. Math. Anal. Appl. 395, 19–29 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shieh, C.T., Buterin, S.A., Ignatiev, M.: On Hochstadt–Liebermann theorem for Sturm–Liouville operators. Far East J. Appl. Math. 52, 131–146 (2011)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Shieh, C.T., Yurko, V.A.: Inverse nodal and inverse spectral problems for discontinuous boundary value problems. J. Math. Anal. Appl. 347, 266–272 (2008)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yang, C.F., Zettl, A.: Half inverse problems for quadratic pencils of Sturm–Liouville operators. Taiwan. J. Math. 16, 1829–1846 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran
  2. 2.Department of MathematicsJouybar Branch, Islamic Azad UniversityJouybarIran
  3. 3.Department of Mathematics, Faculty of ScienceFirat UniversityElazigTurkey

Personalised recommendations