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

Authors

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.

Author Biographies

Mehmet Açil, Department of Mathematics, Van Yüzüncü Yıl University, Turkey

is currently an Assistant Professor of Applied Mathematics at Van Yüzüncü Yıl University (Van YYU), Türkiye since 2019. He obtained his PhD degree in applied mathematics from Van YYU in 2018 specialization in inverse Sturm-Liouville problems and the master degree in applied mathematics from Manisa Celal Bayar University in 2013. His research interests include inverse Sturm-Liouville problems, potential theory and Lie symmetries.

Ali Konuralp, Department of Mathematics, Faculty of Arts and Sciences, Celal Bayar University, Muradiye Campus, Yunus Emre, 45140 Manisa, Turkey

is currently an Associate Professor at Department of Mathematics, Manisa Celal Bayar University in Türkiye, since 2016. He received the master and Ph.D degree in applied mathematics at MCBU. His research interests include in numerical analysis of differential equations and fractional differential equations.

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

2021-05-12

How to Cite

Açil, M., & Konuralp, A. (2021). Reconstruction of potential function in inverse Sturm-Liouville problem via partial data. 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

Research Articles