Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms

Abstract

The purpose of this present paper is to obtain the hypergeometric and discrete fractional solutions of the radial Schrödinger equation by using the nabla discrete fractional calculus operator.

Keywords

• Fractional calculus,discrete fractional calculus,Leibniz rule,radial Schrödinger equation

References

1. Abu-Saris, R. and Al-Mdallal, Q., On the asymptotic stability of linear system of fractional-order difference equations, Fract. Calc. Appl. Anal. 16(3) (2013), 613-629.
2. Acar, N. and Atici, F. M., Exponential functions of discrete fractional calculus, Appl. Anal. Discrete Math. 7 (2013), 343-353.
3. Atici, F. M. and Eloe, P. W., Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. 3 (2009), 1-12.
4. Atici, F. M. and Uyanik, M., Analysis of discrete fractional operators, Appl. Anal. Discrete Math. 9(1) (2015), 139-149.
5. Baoguo, J., Erbe, L. and Peterson, A., Convexity for nabla and delta fractional differences, J. Difference Equ. Appl. 21(4) (2015), 360-373.
6. Belgacem, F. B. M., Sumudu Transform Applications to Bessel Functions and Equations, Appl. Math. Sci. 4(74) (2010), 3665-3686.
7. Benci, V. and D'Aprile, T., The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations 184(1) (2002), 109-138.
8. Chaurasia, V. B. L., Dubey, R. S. and Belgacem, F. B. M., Fractional radial diffusion equation analytical solution via Hankel and Sumudu transforms, Mathematics in Engineering, Science and Aerospace 3(2) (2012), 179-188.

9. Cheng, Y.-F. and Dai, T.-Q., 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) (2007), 274.
10. Chen, Y. and Tang, X., The difference between a class of discrete fractional and integer order boundary value problems, Commun. Nonlinear Sci. Numer. Simulat. 19 (2014), 4057-4067.
11. Goswami, P. and Belgacem, F. B. M., Fractional differential equation solutions through a Sumudu rational, Nonlinear Sci. 19(4) (2012), 591-598.
12. Gupta, V. G., Sharma, B. and Belgacem, F. B. M., On the solutions of generalized fractional kinetic equations, Appl. Math. Sci. 5(19) (2011), 899-910.
13. He, Y. and Hou, C., Existence of solutions for discrete fractional boundary value problems with p-Laplacian operator, J. Math. Res. Appl. 34 (2014), 197-208.
14. Holmer, J. and Roudenko, S., A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Commun. Math. Phys. 282 (2008), 435-467.
15. Katatbeh, Q. D. and Belgacem, F. B. M., Applications of the Sumudu transform to fractional differential equations, Nonlinear Stud. 18(1) (2011), 99-112.
16. Lv, W., Existence and uniqueness of solutions for a discrete fractional mixed type sum-difference equation boundary value problem, Discrete Dyn. Nat. Soc. 2015 (2015), 1-10. doi: 10.1155/2015/376261.
17. Mohan, J. J., Solutions of perturbed nonlinear nabla fractional difference equations, Novi Sad J. Math. 43 (2013), 125-138.
18. Mohan, J. J., Variation of parameters for nabla fractional difference equations, Novi Sad J. Math. 44(2) (2014), 149-159.
19. Ozturk, O., A study on nabla discrete fractional operator in mass-spring-damper system, New Trends Math. Sci. 4(4) (2016), 137-144.
20. Ozturk, O. and Yilmazer, R., Solutions of the radial Component of the fractional Schrödinger equation using N-fractional calculus operator, Differ. Equ. Dyn. Syst. (2016), 1-9. doi: 10.1007/s12591-016-0308-8.
21. Reunsumrit, J. and Sitthiwirattham, T., On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations, Math. Methods Appl. Sci. 39 (2016), 2737-2751.
22. Tselios, K. and Simos, T. E., Symplectic methods for the numerical solution of the radial Shrödinger equation, J. Math. Chem. 34(1-2) (2003), 83-94.
23. Yilmazer, R. and Ozturk, O., Explicit solutions of singular differential equation by means of fractional calculus operators, Abstr. Appl. Anal. 2013 (2013), 1-6. doi: 10.1155/2013/715258.
24. Yilmazer, R., Inc, M., Tchier, F. and Baleanu, D., Particular solutions of the confluent hypergeometric differential equation by using the nabla fractional calculus operator, Entropy 18(2) (2016), 1-6. doi: 10.3390/e18020049
25. Znojil, M., On exact solutions of the Schrödinger equation, J. Phys. A: Math. Gen. 16(2) (1983), 279.