Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator

Year 2023, Volume: 6 Issue: 3, 142 - 147, 17.09.2023

Yaşar Çakmak

https://doi.org/10.33434/cams.1281434

Abstract

In this paper, we prove an Ambarzumyan-type theorem for a Conformable fractional diffusion operator, i.e. we show that $q(x)$ and $p(x)$ functions are zero if the eigenvalues are the same as the eigenvalues of zero potentials.

Keywords

Ambarzumyan-type theorem, Conformable fractional derivative, Diffusion operator, Inverse problem

References

• [1] V.A. Ambarzumian, Uber eine frage der eigenwerttheorie, Zeitschrift f¨ur Physik, 53 (1929), 690-695.
• [2] H.H. Chern, C.K. Law, H.J. Wang, Extensions of Ambarzumyan’s theorem to general boundary conditions, J. Math. Anal. Appl., 263 (2001), 333-342. Corrigendum: J. Math. Anal. Appl., 309 (2) (2005), 764-768.
• [3] E.M. Harrell, On the extension of Ambarzunyan’s inverse spectral theorem to compact symmetric spaces, Amer. J. Math., 109 (1987), 787-795.
• [4] M. Horvath, On a theorem of Ambarzumyan, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 131 (2001), 899-907.
• [5] M. Kiss, An n-dimensional Ambarzumyan type theorem for Dirac operators, Inverse Problems, 20 (5)(2004), 1593-1597.
• [6] C.F. Yang, X.P. Yang, Some Ambarzumyan-type theorems for Dirac operators, Inverse Problems, 25 (9) (2009), 095012, 13 pages.
• [7] C.F. Yang, Z.Y. Huang, X.P. Yang, Ambarzumyan’s theorems for vectorial Sturm-Liouville systems with coupled boundary conditions, Taiwanese J. Math., 14 (4) (2010), 1429-1437.
• [8] C.L. Shen, On some inverse spectral problems related to the Ambarzumyan problem and the dual string of the string equation, Inverse Problems, 23 (6) (2007), 2417-2436.
• [9] N.V. Kuznetsov, Generalization of a theorem of V.A. Ambarzumyan, Doklady Akademii Nauk SSSR., 146 (1962), 1259- 1262, (in Russian).
• [10] H.H. Chern, C.L. Shen, On the n-dimensional Ambarzumyan’s theorem, Inverse Problems, 13 (1) (1997), 15-18.
• [11] C.F. Yang, X.P. Yang, Ambarzumyan’s theorem with eigenparameter in the boundary conditions, Acta Math. Sci., 31 B(4) (2011), 1561-1568.
• [12] V.A. Yurko, On Ambarzumyan-type theorems, Appl. Math. Lett., 26 (4) (2013), 506-509.
• [13] K. Marton, An n-dimensional Ambarzumyan type theorem for Dirac operators, Inverse Problems, 20 (5) (2004), 1593-1597.
• [14] A.A. Kırac, Ambarzumyan’s theorem for the quasi-periodic boundary conditions, Anal. Math. Phys., 6 (2016), 297-300.
• [15] H. Koyunbakan, D. Lesnic, E.S. Panakhov, Ambarzumyan type theorem for a quadratic Sturm-Liouville operator, Turkish Journal of Sciences and Technology, 8 (1) (2013), 1-5.
• [16] G. Freiling, V.A. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York, 2001.
• [17] E. Yilmaz, H. Koyunbakan, Some Ambarzumyan type theorems for Bessel operator on a finite interval, Differ. Equ. Dyn. Syst., 27 (4) (2019), 553–559.
• [18] R. Khalil, M. Al Horania, A. Yousefa, et al., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
• [19] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.
• [20] A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898.
• [21] M. Abu Hammad, R. Khalil, Abel’s formula and Wronskian for conformable fractional differential equations, International J. Differ. Equ. Appl., 13 (3) (2014), 177-183.
• [22] O.T. Birgani, S. Chandok, N. Dedovic, S. Radenoviç, A note on some recent results of the conformable derivative, Advances in the Theory of Nonlinear Analysis and Its Applications, 3 (1) (2019), 11-17.
• [23] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (3) (2017), 903-917.
• [24] H.W. Zhou, S. Yang, S.Q. Zhang, Conformable derivative approach to anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 491 (2018), 1001-1013.
• [25] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), Article ID 247, 16 pages.
• [26] H. Mortazaasl, A. Jodayree Akbarfam, Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem, Inverse Probl. Sci. Eng., 28 (4) (2020), 524-555.
• [27] B. Keskin, Inverse problems for one dimensional conformable fractional Dirac type integro differential system, Inverse Problems, 36 (6) (2020), 065001.
• [28] E. Yilmaz, T. Gulsen, E.S. Panakhov, Existence results for a conformable type Dirac system on time scales in quantum physics, Appl. Comput. Math., 21 (3) (2022), 279-291.
• [29] I. Adalar, A.S. Ozkan, Inverse problems for a conformable fractional Sturm-Liouville operators, J. Inverse III Posed Probl., 28 (6) (2020), 775-782.
• [30] Y. Çakmak, Inverse nodal problem for a conformable fractional diffusion operator, Inverse Probl. Sci. Eng., 29 (9) (2021), 1308-1322.
• [31] Y. Çakmak, Trace formulae for a conformable fractional diffusion operator, Filomat, 36 (14) (2022), 4665-4674.
• [32] E. Koç, Y. Çakmak, a-integral representation of the solution for a conformable fractional diffusion operator and basic properties of the operator, Cumhuriyet Science Journal, 44 (1) (2023), 170-180.
• [33] B.P. Allahverdiev, H. Tuna, Y. Yalçınkaya, Conformable fractional Sturm-Liouville equation, Math. Methods Appl. Sci., 42 (10) (2019), 3508-3526.
• [34] D. Baleanu, Z.B. Guvenc, J.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
• [35] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
• [36] C.A. Monje, Y. Chen, B.M. Vinagre, et al., Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, London, 2010.
• [37] A. Palfalvi, Efficient solution of a vibration equation involving fractional derivatives, Internat. J. Non-Linear Mech., 45 (2010), 169–175.
• [38] M.F. Silva, J.A.T. Machado, Fractional order PDa joint control of legged robots, J. Vib. Control, 12 (12) (2006), 1483–1501