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Broyden method for inverse non-symmetric Sturm-Liouville problems

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Abstract

In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov’s method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss.

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References

  1. Aceto, L., Ghelardoni, P., Magherini, C.: Boundary Value Methods as an extension of Numerov’s method for Sturm–Liouville eigenvalue estimates. Appl. Numer. Math. 59, 1644–1656 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aceto, L., Ghelardoni, P., Magherini, C.: BVMs for Sturm-Liouville eigenvalue estimates with general boundary conditions. J. Numer. Anal. Ind. Appl. Math. 4, 113–127 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Andrew, A.L., Paine, J.W.: Correction of Numerov’s eigenvalue estimates. Numer. Math. 47, 289–300 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrew, A.L.: Finite and continuous perturbations of matrix eigenvalues. Appl. Math. Lett., 11(1), 47–51 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andrew, A.L.: Asymptotic correction of Numerov’s eigenvalue estimates with natural boundary conditions. J. Comput. Appl. Math. 125, 359–366 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Andrew, A.L.: Asymptotic correction of more Sturm-Liouville eigenvalue estimates. BIT Numer. Math. 43, 485–503 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andrew, A.L.: Numerical solution of inverse Sturm-Liouville problems. ANZIAM J. 45(E), C326–C337 (2004)

    MathSciNet  Google Scholar 

  8. Andrew, A.L.: Numerov’s method for inverse Sturm-Liouville problems. Inverse Probl. 21, 223–238 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Andrew, A.L.: Computing Sturm–Liouville potentials from two spectra. Inverse Probl. 22, 2069–2081 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Berghe, G.V., Daele, M.V.: Exponentially–fitted Numerov methods. J. Comput. Appl. Math. 200, 140–153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Acta Math. 78, 1–96 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  12. Broyden, C.G., Dennis, J.E. Jr., Moré, J.J.: On the local and superlinear convergence of quasi-Newton methods. IMA J. Appl. Math. 12, 223–246 (1973)

    Article  MATH  Google Scholar 

  13. Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon & Breach, Amsterdam (1998)

    Google Scholar 

  14. Chadan, K., Colton, D., Päivärinta, L., Rundell, W.: An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  15. Dennis, J.E. Jr.: On the convergence of Broyden’s method for nonlinear systems of equations. Math. Comput. 25, 559–567 (1971)

    MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gel’fand, I.M., Levitan, B.M.: On the determination of a differential equation from its spectral function. Am. Math. Soc. Transl. 1, 253–305 (1955)

    MathSciNet  MATH  Google Scholar 

  18. Ghelardoni, P.: Approximations of Sturm–Liouville eigenvalues using boundary value methods. Appl. Numer. Math. 23, 311–325 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ghelardoni, P., Magherini, C.: BVMs for computing Sturm-Liouville symmetric potentials. Appl. Math. Comput. 217, 3032–3045 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, New York (1987)

    MATH  Google Scholar 

  21. Kammanee, A., Böckmann, C.: Determination of partially known Sturm–Liouville potentials. Appl. Math. Comput. 204, 928–937 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kammanee, A., Böckmann, C.: Boundary value method for inverse Sturm-Liouville problems. Appl. Math. Comput. 214, 342–352 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ledoux, V., Daele, M.V., Berghe, G.V.: MATSLISE: A Matlab package for the numerical solution of Sturm-Liouville and Schrödinger equations. ACM Trans. Math. Softw. 31, 532–554 (2005). Available at http://users.ugent.be/~vledoux/MATSLISE/

    Article  MATH  Google Scholar 

  24. Lowe, B.D., Pilant, M., Rundell, W.: The recovery of potentials from finite spectral data. SIAM J. Math. Anal. 23, 482–504 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. McLaughlin, J.R.: Analytic methods for recovering coefficients in differential equations from spectral data. SIAM Rev. 28, 53–72 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Academic Press, London (1987)

    MATH  Google Scholar 

  28. Pryce, J.D.: Numerical Solution of Sturm–Liouville Problems. Oxford University Press, New York (1993)

    MATH  Google Scholar 

  29. Rafler, M., Böckmann, C.: Reconstruction methods for inverse Sturm-Liouville problems with discontinuous potentials. Inverse Probl. 23, 933–946 (2007)

    Article  MATH  Google Scholar 

  30. Rundell, W., Sacks, P.E.: Reconstruction techniques for classical inverse Sturm-Liouville problems. Math. Comput. 58, 161–183 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schwetlick, H.: Numerische Lösung nichtlinearer Gleichungen. Oldenbourg, Munich (1979)

    Google Scholar 

  32. Sun, J.-G.: Multiple eigenvalue sensitivity analysis. Linear Algebra Appl. 137/138, 183–211 (1990)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, 14469, Potsdam, Germany

    Christine Böckmann

  2. Department of Mathematics, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, 90112, Thailand

    Athassawat Kammanee

Authors
  1. Christine Böckmann
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  2. Athassawat Kammanee
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Correspondence to Christine Böckmann.

Additional information

Communicated by Per Lötstedt.

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Böckmann, C., Kammanee, A. Broyden method for inverse non-symmetric Sturm-Liouville problems. Bit Numer Math 51, 513–528 (2011). https://doi.org/10.1007/s10543-011-0317-5

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  • DOI: https://doi.org/10.1007/s10543-011-0317-5

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