Fractional approach for Dirac operator involving M-truncated derivative

Year 2024, Volume: 73 Issue: 1, 259 - 273, 16.03.2024

Ahu Ercan

https://doi.org/10.31801/cfsuasmas.1316623

Abstract

In this study, we examine the basic spectral information for systems governed by the Dirac equation with distinct boundary conditions, utilizing a modified form of local derivatives known as M-truncated derivative (MTD). The spectral information discussed includes the representation of solutions in the form of integral equations, the asymptotics vector-valued eigenfunctions and eigenvalues, and their normalized forms, all within the context of the MTD method that incorporates truncated Mittag-Leffler functions. This type of MTD provides the features of integer-order operator theory. Also, by virtue of the parameters $\alpha $ and $\gamma$, we analyze and compare the solutions with graphs in terms of different potentials, different eigenvalues and different orders. Thus, the aim of this article is to consider spectral structure of Dirac system in frame of M-truncated derivative by proping with visual analysis.

Keywords

M-truncated derivative, Dirac operator, spectral data, visual results

References

• Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66. https://doi.org/10.1016/j.cam.2014.10.016
• Allahverdiev, B. P., Tuna, H. One-dimensional conformable fractional Dirac system, Bol. Soc. Mat. Mex., 26(1)(2020), 121-146. https://doi.org/10.1007/s40590-019-00235-5
• Allahverdiev, B. P., Tuna, H., Spectral expansion for singular conformable fractional Dirac systems, Rend. Circ. Mat. Palermo, 69(3) (2020), 1359–1372. https://doi.org/10.1007/s12215-019-00476-3
• Allahverdiev, B. P., Tuna, H., Regular fractional Dirac type systems, Facta Univ. Ser. Math. Inform., 36(3) (2021), 489-499. https://doi.org/10.22190/FUMI200318036A
• Al-Refai, M., Abdeljawad, T., Fundamental results of conformable Sturm-Liouville eigenvalue problems, Complexity, (2017), 1-7. https://doi.org/10.1155/2017/3720471
• Anderson, D. R., Ulness, D. J., Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10(2) (2015), 109-137.
• Anderson, D. R., Ulness, D. J., Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phys., 56(6) (2015), 063502. https://doi.org/10.1063/1.4922018.
• Atangana, A., Baleanu, D., Alsaedi, A., New properties of conformable derivative, Open Math., 13(1) (2015), 889-898. https://doi.org/10.1515/math-2015-0081
• Baleanu, D., Jarad, F., U˘gurlu, E., Singular conformable sequential differential equations with distributional potentials. Quaest. Math., 42(3) (2019), 277-287. https://doi.org/10.2989/16073606.2018.1445134
• Bjorken, J. D., Drell, S. D., Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964.
• Ercan, A., Panakhov, E. S., Stability of the spectral problem for Dirac operators, Aip Conf. Proc., 1738 (2016), 290010.
• Ercan, A., On the fractional Dirac systems with non-singular operators, Thermal Sci., 23(6) (2019), 2159-2168. https://doi.org/10.2298/TSCI190810405E
• Ercan, A., Bas, E., Regular spectral problem for conformable Dirac system with simulation analysis, J. Interdiscip. Math., 24(6) (2021), 1497-1514. https://doi.org/10.1080/09720502.2020.1827507
• Greiner, W., Miller, B., Rafelski, J., Quantum Electrodynamics of Strong Fields, Springer, Berlin, 1985.
• Greiner, W., Relativistic Quantum Mechanics: Wave Equations, Springer, Berlin, 1994.
• Hammad, M. A., Khalil, R., Abel’s formula and wronskian for conformable fractional differential equations, Int. J. Differ. Equ. Appl., 13(3) (2014), 177-183. http://dx.doi.org/10.12732/ijdea.v13i3.1753
• Horani, M. A., Hammad, M. A., Khalil, R., Variation of parameters for local fractional nonhomogenous linear-differential equations, J. Math. Comput. Sci., 16 (2016), 147-153.
• Jarad, F., Uğurlu, E., Abdeljawad, T., Baleanu, D., On a new class of fractional operators, Adv. Difference Equ., 247 (2017), 16. https://doi.org/10.1186/s13662-017-1306-z
• Katugampola, U. N., A new fractional derivative with classical properties, arXiv preprint, (2014), arXiv:1410.6535v2.
• Khalil, R., Horani, M. A., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. https://doi.org/10.1016/j.cam.2014.01.002
• Levitan, B. M., Sargsjan, I. S., Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, American Mathematical Society, Providence, R.I., 1975.
• Levitan, B. M., Sargsjan, I. S., Sturm-Liouville and Dirac Operators, Kluwer Academic Publishers Group, Dordrecht, 1991.
• Mamedov, K. R., Akcay, O., Inverse problem for a class of Dirac operators by the Weyl function, Dynam. Systems Appl., 26(1) (2017), 183-195.
• Ozarslan, R., Bas, E., Baleanu, D., Acay, B., Fractional physical problems including windinfluenced projectile motion with Mittag-Leffler kernel, AIMS Math., 5(1) (2019), 467-481. https://doi.org/10.3934/math.2020031
• Vanterler da C. Sousa J., Capelas de Oliveira, E., A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl., 16(1) (2018), 83-96.
• Vanterler da C. Sousa J., Capelas de Oliveira, E., Leibniz type rule: ψ- Hilfer fractional operator, Nonlinear Sci. Numer. Simul., 77 (2019), 305–311. https://doi.org/10.1016/j.cnsns.2019.05.003
• Vanterler da C. Sousa J., Capelas de Oliveira, E., Mittag–Leffler functions and the truncated V-fractional derivative, Mediterr. J. Math., 14(6) (2017), 244. https://doi.org/10.1007/s00009-017-1046-z
• Yalçınkaya, Y., Some fractional Dirac systems, Turkish J. Math., 47(1) (2023), 110-122. https://doi.org/10.55730/1300-0098.3349
• Yusuf, A., Inc, M., Aliyu, A. I., Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel, Chaos, 28(12) (2018), 123121, 11. https://doi.org/10.1063/1.5074084
• Yusuf, A., Sulaiman, T. A., Mirzazadeh, M., Hosseini, K., M-truncated optical solitons to a nonlinear Schr¨odinger equation describing the pulse propagation through a two-mode optical fiber, Opt. Quant. Electron, 53(10) (2021), 558. https://doi.org/10.1007/s11082-021-03221-2
• Yusuf, A., Sulaiman, T. A., Inc, M., Abdel-Khalek, S., Mahmoud, K. H., M-truncated optical soliton and their characteristics to a nonlinear equation governing the certain instabilities of modulated wave trains, AIMS Math., 6(9) (2021), 9207–9221. https://doi.org/10.3934/math.2021535