The Parametric Generalized Fractional Nikiforov-Uvarov Method and Its Applications

  • M. Abu-Shady Department of Mathematics and Computer Sciences, Faculty of Science, Menoufia University, Egypt
  • H.M. Fath-Allah Higher Institute of Engineering and Technology, Menoufia, Egypt
Keywords: Nonrelativistic models, Generalized fractional derivative, Molecular physics, Hadronic physics


   By using generalized fractional derivative, the parametric generalized fractional Nikiforov-Uvarov (NU) method is introduced. The second-order parametric generalized differential equation is exactly solved in the fractional form. The obtained results are applied on the extended Cornell potential, the pesudoharmonic potential, the Mie potential, the Kratzer-Fues potential, the harmonic oscillator potential, the Morse potential, the Woods-Saxon potential, the Hulthen potential, the deformed Rosen-Morse potential and the P schl-Teller potential which play an important role in the fields of molecular and atomic physics. The special of classical cases are obtained from the fractional cases at  which are agreement with recent works.


Download data is not yet available.


R. Hilfer, editor, Applications of Fractional Calculus in Physics, (World Scientific, Singapore, 2000).

M. Dalir, and M. Bashour, Appl. Math. Sci. 4(21), 1021–540 (2010).

G. Wang, and A.-M. Wazwaz, “On the modified Gardner type equation and its time fractional form,” Chaos, Solitons & Fractals 155, 111694 (2022).

G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, 51(9-10), 1367-1376 (2006).

A. Atangana and A. Secer, “A note on fractional order derivatives and table of fractional derivatives of some special functions,” Abstract and Applied Analysis, 2013, 279681 (2013).

K.S. Miller, and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, (Wiley, New York, 1993).

M. Caputo, “Linear models of dissipation whose Q is almost frequency independent – II,” Geophysical Journal International, 13(5), 529-539 (1967). ‏

M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” Journal of computational and applied mathematics, 264, 65-70‏ (2014).

T. Abdeljawad, “On conformable fractional calculus,” Journal of computational and Applied Mathematics, 279, 57-66‏ (2015).

M. Abu-Shady, and M.K.A. Kaabar, “A generalized definition of the fractional derivative with applications,” Mathematical Problems in Engineering, 2021, 9444803 (2021).‏

M. Abu-Shady, T.A. Abdel-Karim, and E.M. Khokha, “Exact Solution of the N-dimensional Radial Schrödinger Equation via Laplace Transformation Method with the Generalized Cornell Potential,” Journal of Quantum Physics, 45, 587 (2018).

M. Abu-Shady, and E. M. Khokha. “Heavy-light mesons in the nonrelativistic quark model using laplace transformation method,” Advances in high energy Physics, 2018, 7032041 (2018).‏

M. Abu-Shady, H.M. Mansour, and A.I. Ahmadov, “Dissociation of quarkonium in hot and dense media in an anisotropic plasma in the nonrelativistic quark model,” Advances in High Energy Physics, 2019, 4785615‏ (2019).

M. Abu-Shady, and H.M. Fath-Allah. “Melting of quarkonium in an anisotropic hot QCD medium in the presence of a generalized Debye screening mass and Nikiforov–Uvarov’s method,” International Journal of Modern Physics A, 35(21), 2050110 (2020).‏

M. Abu-Shady, and A.N. Ikot, “Dissociation of nucleon and heavy baryon in an anisotropic hot and dense QCD medium using Nikiforov–Uvarov method,” The Eurpean Physical Journal Plus, 135(5), 1-17 (2020).

M. Abu-Shady, “Multidimensional Schrödinger Equation and Spectral Properties of Heavy-Quarkonium Mesons at Finite Temperature,” Advances in Mathematical Physics, 2016, 4935940 (2016).‏‏

M. Abu-Shady, “Heavy quarkonia and Bc-mesons in the cornell potential with harmonic oscillator potential in the N-dimensional Schrodinger equation,” International Journal of Applied Mathematics and Theoretical Physics, 2, 16-20 (2015).

M. Abu-Shady, “Quarkonium masses in a hot QCD medium using conformable fractional of the Nikiforov–Uvarov method,” International Journal of Modern Physics A, 34(31), 1950201 (2019).

M. Abu-Shady, and E.P. Inyang, “Heavy-Light Meson Masses in the Framework of Trigonometric Rosen-Morse Potential Using the Generalized Fractional Derivative,” East European Journal of Physics, 4, 80-86 (2022).‏

M. Abu-Shady, T.A. Abdel-Karim, and E.M. Khokha, “The Generalized Fractional NU Method for the Diatomic Molecules in the Deng-Fan Model,” The European Physical Journal D, 67, 159 (2022).

M. Abu-Shady, and Sh.Y. Ezz-Alarab, “Conformable Fractional of the Analytical Exact Iteration Method for Heavy Quarkonium Masses Spectra,” Few-Body Systems, 62(2), 13 (2021). ‏

M. Abu-shady, A.I. Ahmadov, H.M. Fath-Allah, and V.H. Badalov, “Spectra of heavy quarkonia in a magnetized-hot medium in the framework of fractional non-relativistic quark model,” Journal of Theoretical and Applied Physics, 16(3), 1-8 (2022).

M. Abu-shady, and H.M. Fath-Allah, “Masses of Single, Double, and Triple Heavy Baryons in the Hyper-Central Quark Model by Using GF-AEIM,” Advances in High Energy Physics, 2022, 4539308 (2022).‏

H. Karayer, D. Demirhan, and F. Büyükkılıç, “Conformable fractional Nikiforov-Uvarov method,” Communications in Theoretical Physics, 66(1), 12 (2016).‏

C. Tezcan, and R. Sever, “A general approach for the exact solution of the Schrödinger equation,” International Journal of Theoretical Physics, 48(2), 337-350 (2009).‏

A. Berkdemir, C. Berkdemir, and R. Sever, “Eigenvalues and eigenfunctions of Woods–Saxon potential in PT-symmetric quantum mechanics,” Modern Physics Letters A, 21(27), 2087-2097 (2006).‏

M.M. Hammad, et al. “Analytical study of conformable fractional Bohr Hamiltonian with Kratzer potential,” Nuclear Physics A, 1015, 122307 (2021). ‏

A.F. Nikiforov, and V.B. Uvarov, Special Functions of Mathematical Physics, (Birkhäuser, Basel, 1988).

V. Kumar, R.M. Singh, S.B. Bhardwaj, and F. Chand, “Spectrum Analysis of Mesons using Nikiforov-Uvarov Functional Analysis Method,” Materials Research Proceedings, 22, 7-12 (2022).‏

S. Ikhdair, and R. Sever, “Exact polynomial eigensolutions of the Schrödinger equation for the pseudoharmonic potential,” Journal of Molecular Structure: THEOCHEM, 806(1-3), 155-158‏ (2007).

R. Sever, C. Tezcan, M. Aktaş, and Ö. Yeşiltaş, “Exact solution of Schrodinger equation for Pseudoharmonic potential,” Journal of Mathematical Chemistry, 43, 845–851 (2008).

R. Sever, M. Bucurgat, C. Tezcan, and Ö. Yeşiltaş, “Bound state solution of the Schrodinger equation for Mie potential,” Journal of Mathematical Chemistry, 34, 749-755 (2007).

Sß. Erkoç, and R. Sever, “1/N expansion for a Mie-type potential,” Physical Review D, 33(2), 588 (1986).‏

G.C. Maitland, M. Rigby, E.B. Smith, and W.A. Wakeham, Intermolecular Forces, (Oxford Univ. Press, Oxford, 1987).

M.R. Pahlavani, editor, Theoretical Concepts of Quantum Mechanics, (InTech, 2012).

P.M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels,” Physical review, 34(1), 57 (1929).

U.S. Okorie, A.N. Ikot, G.J. Rampho, P.O. Amadi, and H.Y. Abdullah, “Analytical solutions of fractional Schrödinger equation and thermal properties of Morse potential for some diatomic molecules,” Modern Physics Letters A, 36(07), 2150041 (2021). ‏

C. Berkdemir, A. Berkdemir, and R. Sever, “Shape-invariance approach and Hamiltonian hierarchy method on the Woods-Saxon potential for ℓ≠ 0 states,” Journal of mathematical chemistry, 43(3), 944-954 (2008).‏

B. Gönül, and K. Köksal, “Solutions for a generalized Woods–Saxon potential,” Physica Scripta, 76(5), 565 (2007).‏

C.O. Edet, and P.O. Okoi, “Any l-state solutions of the Schrödinger equation for q-deformed Hulthen plus generalized inverse quadratic Yukawa potential in arbitrary dimensions,” Revista mexicana de Física, 65(4), 333-344‏ (2019).

H. Eğrifes, D. Demirhan, and F. Büyükkiliç, “Exact solutions of the Schrödinger equation for two “deformed” hyperbolic molecular potentials,” Physica Scripta, 60(3), 195 (1999). ‏

S.-H. Dong, W.-C. Qiang, and J. Garcia-Ravelo, “Analytical approximations to the Schrödinger equation for a second Pöschl–Teller-like potential with centrifugal term,” International Journal of Modern Physics A, 23(10), 1537-1544 (2008).‏

M. Şimęk, and Z. Yalçin, “Generalized Pöschl-Teller potential,” Journal of Mathematical Chemistry, 16(1), 211-215 (1994).‏

How to Cite
Abu-Shady, M., & Fath-Allah, H. (2023). The Parametric Generalized Fractional Nikiforov-Uvarov Method and Its Applications. East European Journal of Physics, (3), 248-262.