# Reconstruction of potential function in inverse Sturm-Liouville problem via partial data

## DOI:

https://doi.org/10.11121/ijocta.01.2021.001090## Keywords:

Sturm-Liouville theory, Numerical approximation of eigenvalues and of other parts of the spectrum, optimization## Abstract

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of R\"{o}hrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the $j$th elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively.

## References

Hald, O.H. (1978). Sturm-Liouville Problem and the Rayleigh-Ritz Method. Math. Comp., 32, 687–705.

Paine, J. (1984). A Numerical method for the Inverse Sturm-Liouville problem. SIAM J. Sci. Stat.Comput., 5, 149–156.

Sacks, P.E. (1988). An iterative method for the inverse Dirichlet problem. Inverse Problems, 4, 1055–1069.

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

Rundell, W., Sacks, P.E. (1992). Reconstruction Techniques for Classical Inverse Sturm Liouville Problems. Math. Comp., 58, 161–183.

Neher, M. (1994). Enclosing solutions of an inverse Sturm-Liouville problem with finite data. Computing, 53, 379–395.

Fabiano, R.H., Knobel, R., & Lowe, B.D. (1995). A finite-difference algorithm for an Sturm-Liouville problem. IMA J. Num. Anal., 15 , 75–88.

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

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

Andrew, A.L. (2011). Finite difference methods for half inverse Sturm-Liouville problems. App. Math. and Comp., 218 , 445– 457.

Brown, B.M., Samko, V.S., Knowles, I.W., Marletta, M. (2003). Inverse spectral problem for the Sturm-Liouville equation. Inverse Problems, 19, 235–252.

Röhrl, N. (2005). A least-squares Functional for solving inverse Sturm-Liouville problems. Inverse Problems, 21, 2009–2017.

Röhrl, N. (2006). Recovering boundary conditions in inverse Sturm-Liouville problems. Recent advances in differential equations and Mathematical physics, Contemp. Math., Amer. Math. Soc., Providence, RI, 412, 263–270.

Rafler, M., B¨ockmann, C. (2007). Reconstruction method for inverse Sturm-Liouville problems with discontinuous potentials. Inverse Problems, 23, 933–946.

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

Ghelardoni, P., Magherini,C. (2010). BVMs for computing Sturm-Liouville symmetric potentials. App. Math. Comp., 217, 3032– 3045.

Gao, Q., Huang, Z., Cheng, X. (2015). A finite difference method for an inverse Sturm- Liouville problem in impedance form. Numer. Algor., 70, 669–690.

Tuz, M. (2017). Boundary values for an eigenvalue problem with a singular potential. An International Journal of Optimization and Control: Theories & Applications, 7(3) , 293–300.

McLaughlin, J.R., Rundell, W. (1987). A uniqueness theorem for an inverse Sturm- Liouville problem. Math. Phys., 28 , 1471– 1472.

Levinson, N. (1949). The Inverse Sturm- Liouville Problem. Mat. Tideskr. B., 25 , 25– 30.

Pöschel, J., Trubowitz, E. (1987). Inverse Spectral Theory. Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 130, x+192 pp, ISBN: 0-12-563040-9.

Polak, E. (1997). Optimization. Algorithms and consistent approximations. Applied Mathematical Sciences, Springer- Verlag, New York 124, xx+779 pp, ISBN: 0-387-94971-2 297–317.

Hoschtadt, H. (1973). The inverse Sturm- Liouville Problem. Commun. Pure Appl. Math., 26, 715–729.

Squire, J. (2013). Eigenvalue Differential Equation Solver. http://library.wolfram.com/infocenter/MathSource/8762/#downloads.

Al-Mdallal, Q.M., Al-Refai, M., Syam, M., & Al-Srihin, M.K. (2018). Theoretical and computational perspectives on the eigenvalues of fourth-order fractional Sturm–Liouville problem. International Journal of Computer Mathematics 95(8) , 1548–1564.

Mert, R., Abdeljawad, T., & Peterson, A. (2018). A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete and Continuous Dynamical Systems Series S, 1–17.

## Downloads

## Published

## How to Cite

*An International Journal of Optimization and Control: Theories &Amp; Applications (IJOCTA)*,

*11*(2), 186–198. https://doi.org/10.11121/ijocta.01.2021.001090

## Issue

## Section

## License

Copyright (c) 2021 Mehmet Açil, Ali Konuralp

This work is licensed under a Creative Commons Attribution 4.0 International License.

Articles published in IJOCTA are made freely available online immediately upon publication, without subscription barriers to access. All articles published in this journal are licensed under the Creative Commons Attribution 4.0 International License (click here to read the full-text legal code). This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.

Under the Creative Commons Attribution 4.0 International License, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles in IJOCTA, so long as the original authors and source are credited.

**The readers are free to:**

**Share**— copy and redistribute the material in any medium or format**Adapt**— remix, transform, and build upon the material- for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.

**under the following terms:**

**Attribution**— You must give**appropriate credit**, provide a link to the license, and**indicate if changes were made**. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.

**No additional restrictions**— You may not apply legal terms or**technological measures**that legally restrict others from doing anything the license permits.

This work is licensed under a Creative Commons Attribution 4.0 International License.