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

Year 2019, Volume: 68 Issue: 1, 833 - 839, 01.02.2019
https://doi.org/10.31801/cfsuasmas.481600

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.

References

  • 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.
  • Acar, N. and Atici, F. M., Exponential functions of discrete fractional calculus, Appl. Anal. Discrete Math. 7 (2013), 343-353.
  • Atici, F. M. and Eloe, P. W., Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. 3 (2009), 1-12.
  • Atici, F. M. and Uyanik, M., Analysis of discrete fractional operators, Appl. Anal. Discrete Math. 9(1) (2015), 139-149.
  • Baoguo, J., Erbe, L. and Peterson, A., Convexity for nabla and delta fractional differences, J. Difference Equ. Appl. 21(4) (2015), 360-373.
  • Belgacem, F. B. M., Sumudu Transform Applications to Bessel Functions and Equations, Appl. Math. Sci. 4(74) (2010), 3665-3686.
  • 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.
  • 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.
  • 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.
  • 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.
  • Goswami, P. and Belgacem, F. B. M., Fractional differential equation solutions through a Sumudu rational, Nonlinear Sci. 19(4) (2012), 591-598.
  • 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.
  • 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.
  • 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.
  • Katatbeh, Q. D. and Belgacem, F. B. M., Applications of the Sumudu transform to fractional differential equations, Nonlinear Stud. 18(1) (2011), 99-112.
  • 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.
  • Mohan, J. J., Solutions of perturbed nonlinear nabla fractional difference equations, Novi Sad J. Math. 43 (2013), 125-138.
  • Mohan, J. J., Variation of parameters for nabla fractional difference equations, Novi Sad J. Math. 44(2) (2014), 149-159.
  • Ozturk, O., A study on nabla discrete fractional operator in mass-spring-damper system, New Trends Math. Sci. 4(4) (2016), 137-144.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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
  • Znojil, M., On exact solutions of the Schrödinger equation, J. Phys. A: Math. Gen. 16(2) (1983), 279.
Year 2019, Volume: 68 Issue: 1, 833 - 839, 01.02.2019
https://doi.org/10.31801/cfsuasmas.481600

Abstract

References

  • 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.
  • Acar, N. and Atici, F. M., Exponential functions of discrete fractional calculus, Appl. Anal. Discrete Math. 7 (2013), 343-353.
  • Atici, F. M. and Eloe, P. W., Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. 3 (2009), 1-12.
  • Atici, F. M. and Uyanik, M., Analysis of discrete fractional operators, Appl. Anal. Discrete Math. 9(1) (2015), 139-149.
  • Baoguo, J., Erbe, L. and Peterson, A., Convexity for nabla and delta fractional differences, J. Difference Equ. Appl. 21(4) (2015), 360-373.
  • Belgacem, F. B. M., Sumudu Transform Applications to Bessel Functions and Equations, Appl. Math. Sci. 4(74) (2010), 3665-3686.
  • 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.
  • 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.
  • 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.
  • 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.
  • Goswami, P. and Belgacem, F. B. M., Fractional differential equation solutions through a Sumudu rational, Nonlinear Sci. 19(4) (2012), 591-598.
  • 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.
  • 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.
  • 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.
  • Katatbeh, Q. D. and Belgacem, F. B. M., Applications of the Sumudu transform to fractional differential equations, Nonlinear Stud. 18(1) (2011), 99-112.
  • 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.
  • Mohan, J. J., Solutions of perturbed nonlinear nabla fractional difference equations, Novi Sad J. Math. 43 (2013), 125-138.
  • Mohan, J. J., Variation of parameters for nabla fractional difference equations, Novi Sad J. Math. 44(2) (2014), 149-159.
  • Ozturk, O., A study on nabla discrete fractional operator in mass-spring-damper system, New Trends Math. Sci. 4(4) (2016), 137-144.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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
  • Znojil, M., On exact solutions of the Schrödinger equation, J. Phys. A: Math. Gen. 16(2) (1983), 279.
There are 25 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Okkes Ozturk This is me 0000-0002-2655-519X

Resat Yilmazer 0000-0002-5059-3882

Publication Date February 1, 2019
Submission Date October 26, 2017
Acceptance Date May 2, 2018
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Ozturk, O., & Yilmazer, R. (2019). Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 833-839. https://doi.org/10.31801/cfsuasmas.481600
AMA Ozturk O, Yilmazer R. Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):833-839. doi:10.31801/cfsuasmas.481600
Chicago Ozturk, Okkes, and Resat Yilmazer. “Solutions of the Radial Schrödinger Equation in Hypergeometric and Discrete Fractional Forms”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 833-39. https://doi.org/10.31801/cfsuasmas.481600.
EndNote Ozturk O, Yilmazer R (February 1, 2019) Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 833–839.
IEEE O. Ozturk and R. Yilmazer, “Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 833–839, 2019, doi: 10.31801/cfsuasmas.481600.
ISNAD Ozturk, Okkes - Yilmazer, Resat. “Solutions of the Radial Schrödinger Equation in Hypergeometric and Discrete Fractional Forms”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 833-839. https://doi.org/10.31801/cfsuasmas.481600.
JAMA Ozturk O, Yilmazer R. Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:833–839.
MLA Ozturk, Okkes and Resat Yilmazer. “Solutions of the Radial Schrödinger Equation in Hypergeometric and Discrete Fractional Forms”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 833-9, doi:10.31801/cfsuasmas.481600.
Vancouver Ozturk O, Yilmazer R. Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):833-9.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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