Inverse Sturm-Liouville problem with conformable derivative and transmission conditions

Yıl 2023, Cilt: 52 Sayı: 3, 753 - 767, 30.05.2023

Mohammad Shahriari Hanif Mirzaei

https://doi.org/10.15672/hujms.1080599

Cited By: 2

Öz

In this paper, we study the inverse problem for Sturm-Liouville problem with conformable fractional differential operators of order $\alpha$, $0.5 < \alpha\leq 1$ and finite number of interior discontinuous conditions. For this aim first, the asymptotic formulas for solutions, eigenvalues and eigenfunctions of the problem are calculated. Then some uniqueness theorems for proposed inverse eigenvalue problem are proved. Finally, the Hald's theorem for conformable Sturm-Liouville problem is developed.

Anahtar Kelimeler

Inverse second order differential problem, conformable fractional derivative, transmission conditions, Weyl function, asymptotic solutions

Kaynakça

• [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279, 57-66, 2015.
• [2] B.P. Allahverdiev, H. Tuna and Y. Yalcinkaya, Spectral expansion for singular conformable Sturm-Liouville problem, Math. Commun. 25 (2), 237-252, 2020.
• [3] J.B. Conway, Functions of One Complex Variable, Springer-Verlag, New York, 1995.
• [4] Y. Çakmak, Inverse nodal problem for a conformable fractional diffusion operator, Inverse Probl. Sci. Eng. 29 (9), 1308-1322, 2021.
• [5] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010.
• [6] G.M.L. Gladwell, Inverse problem in vibration, Kluwer Academic Publishers, New York, 2004.
• [7] T. Gulshen, E. Yilmaz and H. Kemaloglu, Conformable fractional Sturm-Liouville equation and some existence results on time scales, Turk. J. Math. 42 (3), 1348-1360, 2018.
• [8] O.H. Hald, Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math. 37 (5), 539-577, 1984.
• [9] H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (4), 676-680, 1987.
• [10] B. Jin and W. Rundell, An inverse Sturm-Liouville problem with a fractional derivative, J. Comput. Phys. 231 (14), 4954-4966, 2012.
• [11] R. Khalil, M. Al Horani and A. Yousef, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65-70, 2014.
• [12] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations,North-Holland Mathematics Studies, Elsevier Science, 2006.
• [13] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer- Verlag, New York, 1996.
• [14] M. Klimek and O.P. Agrawal, Fractional Sturm-Liouville problem. Comput. Math. Appl. 66 (5), 795-812, 2013.
• [15] B.Y. Levin, Lectures on Entire Functions, Transl. Math. Monographs, American Mathematical Society, 1996.
• [16] H. Mortazaasl and A. Jodayree Akbarfam, Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem, Inverse Probl. Sci. Eng. 28 (4), 524-555, 2020.
• [17] G. Mutlu, Associated functions of non-selfadjoint Sturm-Liouville operator with operator coefficient, TWMS Journal of Applied and Engineering Mathematics 11 (1), 113-121, 2020.
• [18] G. Mutlu and E.K. Arpat, Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis, Hacet. J. Math. Stat. 49 (5), 1686-1694, 2020.
• [19] A.S. Ozkan and İ. Adalar, Inverse problems for a conformable fractional Sturm- Liouville operator, J. Inverse Ill-Posed Probl. 28 (6), 775-782, 2020.
• [20] A. P´alfalvi, Efficient solution of a vibration equation involving fractional derivatives. Int. J. Nonlin. Mech. 45, 169-175, 2010.
• [21] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [22] M. Rivero, J.J. Trujillo and M.P. Velasco, A fractional approach to the Sturm-Liouville problem, Centr. Eur. J. Phys. 11 (10), 1246-1254, 2013.
• [23] A.M. Sedletski, Asymptotic formulas for zeros of functions of MittagLeffler type, Anal. Math. 20 (2), 117-132, 1994.
• [24] M. Shahriari, Inverse Sturm-Liouville problem with eigenparameter dependent boundary and transmission conditions, Azerb. J. Math. 4 (2), 16-30, 2014.
• [25] M. Shahriari, A.J. Akbarfam and G. Teschl, Uniqueness for inverse Sturm-Liouville problems with a finite number of transmission conditions, J. Math. Anal. Appl. 395, 19-29, 2012.
• [26] M. Shahriari, M. Fallahi and F. Shareghi, Reconstruction of the Sturm-Liouville operators with a finite number of tranmission and parameter dependent boundary conditions, Azerb. J. Math. 8 (2), 3-20, 2018.
• [27] G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, Graduate Studies in Mathematics, American Mathematical Society, Rhode Island, 2009.
• [28] C.F. Yang, An Interior inverse problem for discontinuous boundary-value problems, Integral Equations Operator Theory 65, 593-604, 2009.
• [29] C.F. Yang and X.P. Yang, An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions, Appl. Math. Lett. 22, 1315-1319, 2009.
• [30] M. Zayernouri, G. Em Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys. 252, 495-517 2013.