A two-index generalization of conformable operators with potential applications in engineering and physics





Conformable operators, Algebraic Methods, Quantum operators, Sturm Liouville operator


We developed a somewhat novel fractional-order calculus workbench as a certain generalization of the Khalil’s conformable derivative. Although every integer-order derivate can naturally be consistent with fully physical-sense problem’s quotation, this is not the standard scenario of the non-integer-order derivatives, even aiming physics systems’s modelling, solely.We revisited a particular case of the generalized conformable fractional derivative and derived a differential operator, whose properties overcome those of the integer-order derivatives, though preserving its clue advantages.Worthwhile noting, that two-fractional indexes differential operator we are dealing, departs from the single-fractional index framework, which typifies the generalized conformable fractional derivative. This distinction leads to proper mathematical tools, useful in generalizing widely accepted results, with potential applications to fundamental Physics within fractional order calculus. The later seems to be especially appropriate for exercising the Sturm-Liouville eigenvalue problem, as well as the Euler-Lagrange equation and to clarify several operator algebra matters.


Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, (2014), 65-70, https: //doi.org/10.1016/j.cam.2014.01.002.

Katugampola, U. N. "A new fractional derivative with classical properties, e-print." arXiv preprint arXiv:1410.6535 8 (2014), https://arxiv.org/abs/1410.6535v2.

Anderson, D. R., and Ulness, D. J. Properties of the Katugampola fractional derivative with potential application in quantum mechanics. Journal of Mathematical Physics, 56(6), (2015), 063502, https://doi.org/10.1063/1.4922018.

Almeida, R., Guzowska, M., and Odzijewicz, T. A remark on local fractional calculus and ordinary derivatives. Open Mathematics, 14(1), (2016), 1122-1124, https://doi.org/ 10.1515/math-2016-0104.

Abdeljawad, T. On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, (2015), 57-66, https://doi.org/10.1016/j.cam.2014.10.016.

Benkhettou, N., Hassani, S., and Torres, D. F. A conformable fractional calculus on arbitrary time scales. Journal of King Saud University-Science, 28(1), (2016), 93-98, https://doi. org/10.1016/j.jksus.2015.05.003.

Atangana, A., Baleanu, D., and Alsaedi, A. New properties of conformable derivative. Open Mathematics, 1(open-issue), (2015), http://hdl.handle.net/20.500.12416/2879.

Chiranjeevi, T., and Biswas, R. K. Closed-form solution of optimal control problem of a fractional order system. Journal of King Saud University-Science, 31(4), (2019), 1042- 1047, https://doi.org/10.1016/j.jksus.2019.02.010.

Al-Refai, M., and Abdeljawad, T. Fundamental results of conformable Sturm-Liouville eigenvalue problems. Complexity, (2017), https://doi.org/10.1155/2017/3720471.

Zhao, D., Pan, X., and Luo, M. A new framework for multivariate general conformable fractional calculus and potential applications. Physica A: Statistical Mechanics and its Applications, 510, (2018), 271-280, https://doi.org/10.1016/j.physa.2018.06.070.

Lazo, M. J., and Torres, D. F. Variational calculus with conformable fractional derivatives. IEEE/CAA Journal of Automatica Sinica, 4(2), (2016), 340-352, https://doi.org/10. 1109/JAS.2016.7510160.

Klimek, M., and Agrawal, O. P. Fractional Sturm–Liouville problem. Computers and Mathematics with Applications, 66(5), (2013), 795 812, https://doi.org/10.1016/j. camwa.2012.12.011.

Alam, M. N., and Li, X. New soliton solutions to the nonlinear complex fractional Schrödinger equation and the conformable time-fractional Klein–Gordon equation with quadratic and cubic nonlinearity. Physica Scripta, 95(4), (2020), 045224, https://doi. org/10.1088/1402-4896/ab6e4e.




How to Cite

E. Reyes-Luis, G. Fernández Anaya, J. Chávez-Carlos, L. Diago-Cisneros, and R. Muñoz Vega, “A two-index generalization of conformable operators with potential applications in engineering and physics”, Rev. Mex. Fís., vol. 67, no. 3 May-Jun, pp. 429–442, May 2021.



07 Gravitation, Mathematical Physics and Field Theory