A On Abu-Shady-Kaabar Fractional Chebyshev Differential Equation of the First Kind

Francisco  Martínez; Prof. Mohammed K. A.  Kaabar

doi:10.5269/bspm.68585

Francisco Martínez Department of Applied Mathematics and Statistics, Technological University of Cartagena, Cartagena 30203, Spain https://orcid.org/0000-0002-3733-1239
Prof. Mohammed K. A. Kaabar Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia

Résumé

The newly proposed Abu-Shady-Kaabar fractional derivative, initiated as generalized form of fractional derivative, is studied in this paper. New results are obtained via this newly proposed definition are investigated. The derivability and integrability of the sum function of fractional power series are studied in this context. Likewise, the solutions’ existence around an ordinary point of a homogeneous sequential linear generalized fractional differential equation of order 2α is analyzed. The series solutions of the generalized fractional Chebyshev differential equation of first kind are obtained. Some interesting properties of the generalized fractional Chebyshev-I polynomials are also included.

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References
[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, in North Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, Netherlands, 2006.
[2] K. S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Hoboken, NJ, USA, 1993.
[3] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative”, Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014.
[4] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo 54 (3), pp. 903–917, 2017.
[5] R. Khalil, M. Al Horani, M. Abu Hammad, “Geometric meaning of conformable derivative via fractional cords”, Journal of Mathematics and Computer Science, vol. 19, no. 4, pp. 241–245, 2019.
[6] A. A. Abdelhakim , “The flaw in the conformable calculus: it is conformable because it is not fractional,” Fractional Calculus and Applied Analysis, vol. 22, no. 2, pp. 242–254, 2019.
[7] M. Abu-Shady, M. K. A. Kaabar, “A Generalized Definition of the Fractional Derivative with Applications”, Mathematical Problems in Engineering, vol. 2021, Article ID 9444803, pp. 1-9, 2021.
[8] F. Martínez, M. K. A. Kaabar, “Novel Investigation of the Abu-Shady–Kaabar Fractional Derivative as a Modeling Tool for Science and Engineering“, Computational and Mathematical Methods in Medicine, vol. 2022, 2022.
[9] T.M. Apostol, Calculus. Volume 1 (2nd edition), John Wiley & Sons, 1991.
[10] M. Spivak, Calculus, Second Edition, Publish or Perisn. Inc, Houston, 1980.
[11] F. Martínez, I. Martínez, M. K. A. Kaabar, S. Paredes, “Some New Results on Conformable Fractional Power Series”, Asia Pacific Journal of Mathematics, vol. 7, no. 31, pp. 1-14, 2020.
[12] M. Al Masalmeh, Series Method to solve conformable fractional Riccati Differential equations, International Journal of Applied Mathematics Research, vol. 6, no. 1, pp. 30-33, 2017.
[13] W.R. Derrick & S.I. Grossman, Elementary Differential Equations with Applications. Addison-Wesley Publising Company, Inc., de Reading, 1981.
[14] M. Abu-Shady, T. A. Abdel-Karim, E. M. Khokha, “The Generalized Fractional NU Method for the Diatomic Molecules in the Deng-Fan Model”, arXiv preprint arXiv:2203.04125.
[15] M. Abu-Shady, M. M. A. Ahmed, N. H. Gerish, “Generalized Fractional of the Extended Nikiforov-Uvaro Method for Heavy Tetraquark Masses Spectra”, Preprint, 2022.
[16] M. Alquran, F. Yousef, F. Alquran, T. A. Sulaiman, and A. Yusuf, “Dual-wave solutions for the quadratic-cubic conformable-Caputo time-fractional Klein-Fock-Gordon equation,” Mathematics and Computers in Simulation, vol. 185, pp. 62–76, 2021.
[17] A. Din, Y. Li, A. Yusuf, A. I. Ali, “Caputo type fractional operator applied to Hepatitis B system,” Fractals, vol. 30, no, 01, 2240023, 2022.
[18] W. Beghami, B. Maayah, S. Bushnaq, O. Abu Arqub, “The Laplace optimized decomposition method for solving systems of partial differential equations of fractional order,” International Journal of Applied and Computational Mathematics, vol. 8, no. 52, 2022.
[19] A. Torres-Hernandez, F. Brambila-Paz, “Sets of fractional operators and numerical estimation of the order of convergence of a family of fractional fixed-point methods,” Fractal and Fractional, vol. 5, no. 4, 240, 2021.
[20] A. Torres-Hernandez, F. Brambila-Paz, R. Montufar-Chaveznava, “Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers,” Applied Mathematics and Computation, vol. 429, 127231, 2022.