The Fractional Calculus Methods for the Radial Schrödinger Equation Given by Some Physical Potentials

  • Okkes OzturkEmail author
Research Article


In this study the solutions of the radial Schrödinger equation are obtained by means of the fractional calculus. Schrödinger equation is used to predict the future behavior of dynamic systems and, plays an important role in Newton’s laws and law of conservation of energy. Here, we assert the different solution methods for the radial Schrödinger equation given by different physical potentials.


Fractional calculus methods Radial Schrödinger equation Fractional differintegral N-fractional calculus operator Ordinary differential equations 

Mathematics Subject Classification

26A33 34A08 


  1. 1.
    Kulish VV, Lage JL (2002) Application of fractional calculus to fluid mechanics. J Fluids Eng 124:803–806CrossRefGoogle Scholar
  2. 2.
    Meral FC, Royston TJ, Magin R (2010) Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul 15:939–945MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. 3.
    Matusu R (2011) Application of fractional order calculus to control theory. Int J Math Models Appl Sci 5(7):1162–1169Google Scholar
  4. 4.
    Magin R, Feng X, Baleanu D (2008) Fractional calculus in NMR. In: Proceedings of the 17th world congress the international federation of automatic control, Seoul–Korea, July 6–11, pp 9613–9618Google Scholar
  5. 5.
    Delavari H, Ghaderi R, Ranjbar NA et al (2010) Adaptive fractional PID controller for robot manipulator. In: Proceedings of FDA’10. The 4th IFAC workshop fractional differentiation and its applications, Badajoz–Spain, October 18–20Google Scholar
  6. 6.
    Yilmazer R, Ozturk O (2013) Explicit solutions of singular differential equation by means of fractional calculus operators. Abstr Appl Anal. MathSciNetGoogle Scholar
  7. 7.
    Podlubny I (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, methods of their solution and some of their applications. Academic Press, CambridgezbMATHGoogle Scholar
  8. 8.
    Tu ST, Chyan DK, Srivastava HM (2001) Some families of ordinary and partial fractional differintegral equations. Integral Transform Spec Funct 11:291–302MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nishimoto K (1997) Kummer’s twenty-four functions and N-fractional calculus. Nonlinear Anal 30:1271–1282MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yilmazer R (2010) N-fractional calculus operator N μ method to a modified hydrogen atom equation. Math Commun 15:489–501MathSciNetzbMATHGoogle Scholar
  11. 11.
    Whittaker ET, Watson GN (1927) A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, 4th edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
  12. 12.
    Fukuhara M (1941) Ordinary differential equations. Iwanami Shoten, TokyoGoogle Scholar
  13. 13.
    Tricomi FG (1954) Funzioni ipergeometriche confluent. Edizioni Cremonese, RomazbMATHGoogle Scholar
  14. 14.
    Watson GN (1944) A treatise on the theory of Bessel functions, 2nd edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
  15. 15.
    Wei Y-C (2015) Some solutions to the fractional and relativistic Schrödinger equations. Int J Theor Math Phys 5(5):87–111Google Scholar
  16. 16.
    Wang P, Huang C (2015) An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J Comput Phys 293:238–251MathSciNetCrossRefzbMATHADSGoogle Scholar
  17. 17.
    Secchi S, Squassina M (2014) Soliton dynamics for fractional Schrödinger equations. Appl Anal 93(8):1702–1729MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ambrosetti A, Ruiz D (2006) Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials. Proc R Soc Edinb Sect A Math 136(5):889–907CrossRefzbMATHGoogle Scholar
  19. 19.
    Znojil M (1983) On exact solutions of the Schrödinger equation. J Phys A Math Gen 16(2):279CrossRefzbMATHADSGoogle Scholar
  20. 20.
    Adam Gh, Ixaru LGr, Corciovei A (1976) A first-order perturbative numerical method for the solution of the radial Schrödinger equation. J Comput Phys 22(1):1–33CrossRefzbMATHADSGoogle Scholar
  21. 21.
    Cheng Y-F, Dai T-Q (2007) Exact solution of the Schrödinger equation for the modified Kratzer potential plus a ring-shaped potential by the Nikiforov–Uvarov method. Phys Scr 75(3):274MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Aygun M, Bayrak O, Boztosun I (2007) Solution of the radial Schrödinger equation for the potential family V(r) = A/r 2 − B/r + Cr κ using the asymptotic iteration method. J Phys B At Mol Opt Phys 40(3):537CrossRefGoogle Scholar
  23. 23.
    Dong S-H, Qiang W-C, Sun G-H et al (2007) Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential. J Phys A Math Theor 40(34):10535CrossRefzbMATHADSGoogle Scholar
  24. 24.
    Thomas RM, Simos TE, Mitsou GV (1996) A family of Numerov-type exponentially fitted predictor–corrector methods for the numerical integration of the radial Schrödinger equation. J Comput Appl Math 67(2):255–270MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ikhdair SM, Sever R (2008) Exact solutions of the modified Kratzer potential plus ring-shaped potential in the D-dimensional Schrödinger equation by Nikiforov–Uvarov method. Int J Mod Phys C 19(2):221CrossRefzbMATHADSGoogle Scholar
  26. 26.
    Ikhdair SM, Sever R (2007) Exact solutions of the radial Schrödinger equation for some physical potentials. Cent Eur J Phys 5(4):516–527Google Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  1. 1.Bitlis Eren UniversityBitlisTurkey

Personalised recommendations