Fractional Dirac systems with Mittag-Leffler kernel

Abstract

In this paper, we study some fractional Dirac-type systems with the Mittag–Leffler kernel. We extend the basic spectral properties of the ordinary Dirac system to the Dirac-type systems with the Mittag–Leffler kernel. First, this problem was handled in a continuous form. The self-adjointness of the operator produced by this system, the reality of its eigenvalues, and the orthogonality of the eigenfunctions have been investigated. Later, similar results were obtained by considering the discrete state.

Keywords

• Fractional differential equations
• Mittag-Leffler kernel
• Dirac systems

References

1. Abdeljawad, T., Baleanu, D., Integration by parts and its applications of a new non-local fractional derivative with Mittag–Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10(3) (2017), 1098-1107. http://dx.doi.org/10.22436/jnsa.010.03.20
2. Abdeljawad, T., On Delta and Nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), Art. ID 406910, 1-12. https://doi.org/10.1155/2013/406910
3. Abdeljawad, T., Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013), 1-16. https://doi.org/10.1186/1687-1847-2013-36
4. Atangana, A., Baleanu, D., New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. https://doi.org/10.2298/TSCI160111018A
5. Abdeljawad, T., Baleanu, D., Discrete fractional differences with nonsingular discrete Mittag–Leffler kernels, Adv. Differ. Equ., 2016(232) (2016), 1-18. https://doi.org/10.1186/s13662-016-0949-5
6. Atici, F. M., Eloe, P. W., Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., Spec. Ed. I , 2009 (2009), 1-12.
7. Bas, E., Ozarslan, R., Baleanu, D., Ercan, A., Comparative simulations for solutions of fractional Sturm–Liouville problems with non-singular operators, Adv. Diff. Equ., 2018(350) (2018). https://doi.org/10.1186/s13662-018-1803-8
8. Ercan, A., On the fractional Dirac systems with non-singular operators, Thermal science, 23(6) 2019, 2159-2168. https://doi.org/10.2298/TSCI190810405E

9. Erdelyi, A., Book Reviews: Higher Transcendental Functions. Vol. III. Based in Part on Notes Left by Harry Bateman. Science, 122, 290, 1955.
10. Goodrich, C., Peterson, A. C., Discrete Fractional Calculus, Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-25562-0
11. Levitan, B. M., Sargsjan, I. S., Sturm–Liouville and Dirac operators. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian).
12. Mert, R., Abdeljawad, T., Peterson, A., A Sturm–Liouville approach for continuous and discrete Mittag–Leffler kernel fractional operators, Discr. Contin. Dynam. System, Series S, 14(7) (2021), 2417-2434. https://doi.org/10.3934/dcdss.2020171
13. Yalçınkaya, Y., Some fractional Dirac systems, Turkish J. Math., 47 (2023), 110–122. doi:10.55730/1300-0098.3349