On the Solution of a Class of Discontinuous Sturm-Liouville Problems

Year 2025, Volume: 8 Issue: 1, 49 - 56, 27.03.2025

Hanlar Reşidoğlu , Döne Karahan , Ufuk Çelik

https://doi.org/10.33434/cams.1605646

https://izlik.org/JA24CK79YA

Abstract

This study examines boundary value problems consisting of a second-order differential equation with discontinuous coefficients and boundary conditions. Asymptotic formulas for the eigenvalues and eigenfunctions of the problem are derived, and an expansion formula is obtained based on the eigenfunctions.

Keywords

Asymptotic formulas , Discontinuous coefficient , Expansion formula , Sturm-Liouville problem

References

• [1] R. S. Anderssen, The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of torsional eigenfrequencies of the Earth, Geophys. J. R. Astr. Soc., 50 (1997), 303-309.
• [2] E. R. Lapwood, T. Usami, Free Oscillations of the Earth, Cambridge Univ. Press, Cambridge, 1981.
• [3] O. H. Hald, Discontinuous inverse eigenvalue problems, Comm. Pure and Appl. Math., 37 (1984), 539-577.
• [4] D. Shepelsky, The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions, Spectral operator theory and related topics: Adv. In Sov. Math., Providence, Amer. Math. Soc., 19 (1994), 209-232.
• [5] M. Kobayashi, A uniqueness for discontinuous inverse Sturm-Liouville problems with symmetric potentials, Inverse Probl., 5 (1985), 767-781.
• [6] G. Freiling, V. Yurko, On inverse Sturm-Liouville Problems and Their Applications, Nova Science Publisher Inc., New York, 2008.
• [7] G. Freiling, V. Yurko, Lectures on Differential Equations of Mathematical Physics-A First Course, Nova Science Publisher Inc., New York, 2008.
• [8] E. N. Akhmedova, On representation of solution of Sturm-Liouville equation with discontinuous coefficients, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerbaijan, 4 (2003), 7-18.
• [9] E. N. Akhmedova, H. M. Huseynova, On eigenvalues and eigenfunctions of one class of Sturm-Liouville with discontinuous coefficients, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 23 (2003), 7-18.
• [10] E. N. Akhmedova, The definition of one class of Sturm-Liouville Operators with discontinuous coefficients by Weyl function, Proceedings of IMM of NAS of Azerbaijan, 30 (2005), 3-8.
• [11] D. Karahan, Kh. R. Mamedov, ˙I. F. Hashimoglu, On main equation for inverse Sturm–Liouville operator with discontinuous coefficient, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., 225 (2023), 73-86.
• [12] Kh. R. Mamedov, D. Karahan, On an inverse problem for Sturm-Liouville equation, EJPAM, 10 (2017), 535-543.
• [13] Kh. R. Mamedov, F. A. Cetinkaya, Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition, Bound. Value Probl., 2013 (2013), 183.
• [14] Ö. Akçay, Kh. R. Mamedov, Inverse spectral problem for Dirac operators by spectral data, Filomat, 31 (2017), 1065-1077.
• [15] R. Bellman, K. L. Kuk, Difference-Differential Equations, M. Mir., 1967.