Research article

Discrete fractional solutions to the effective mass Schrödinger equation by mean of nabla operator

  • Received: 14 October 2019 Accepted: 02 January 2020 Published: 07 January 2020
  • MSC : 97M50, 97N70

  • In the current article, we investigate the second order singular differential equation namely the effective mass Schrödinger equation by means of the fractional nabla operator. We apply some classical transformations in order to reduce the governing equation, and also restrict the difference parameters involved in order to find them values. In order to achieve these important results, certain tools such as the Leibniz rule, the index law, the shift operator, and the power rule are provided in view of the discrete fractional calculus. We use all these mentioned data for two representations of the given model for homogeneous and non-homogeneous instances. The main advantage of the fractional nabla operator is to apply the singular differential equations and transform them into a fractional order model. As a result, we produce some new exact fractional solutions to the present model for a given potential.

    Citation: Karmina K. Ali, Resat Yilmazer. Discrete fractional solutions to the effective mass Schrödinger equation by mean of nabla operator[J]. AIMS Mathematics, 2020, 5(2): 894-903. doi: 10.3934/math.2020061

    Related Papers:

  • In the current article, we investigate the second order singular differential equation namely the effective mass Schrödinger equation by means of the fractional nabla operator. We apply some classical transformations in order to reduce the governing equation, and also restrict the difference parameters involved in order to find them values. In order to achieve these important results, certain tools such as the Leibniz rule, the index law, the shift operator, and the power rule are provided in view of the discrete fractional calculus. We use all these mentioned data for two representations of the given model for homogeneous and non-homogeneous instances. The main advantage of the fractional nabla operator is to apply the singular differential equations and transform them into a fractional order model. As a result, we produce some new exact fractional solutions to the present model for a given potential.


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    [1] P. K. Jha, H. Eleuch, Y. V. Rostovtsev, Analytical solution to position dependent mass Schrödinger equation, J. Mod. Optic., 58 (2011), 652-656. doi: 10.1080/09500340.2011.562617
    [2] H. Eleuch, P. K. Jha, Y. V. Rostovtsev, Analytical solution to position dependent mass for 3D-Schrodinger equation, Math. Sci. Lett., 1 (2012), 1-6. doi: 10.12785/msl/010101
    [3] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2
    [4] J. M. Luttinger, W. Kohn, Motion of electrons and holes in perturbed periodic fields, Phys. Rev., 97 (1955), 869.
    [5] A. de Souza Dutra, C. A. S. Almeida, Exact solvability of potentials with spatially dependent effective masses, Phys. Lett. A, 275 (2000), 25-30. doi: 10.1016/S0375-9601(00)00533-8
    [6] L. Dekar, L. Chetouani, T. F. Hammann, An exactly soluble Schrödinger equation with smooth position-dependent mass, J. Math. Phys., 39 (1998), 2551-2563. doi: 10.1063/1.532407
    [7] L. Dekar, L. Chetouani, T. F. Hammann, Wave function for smooth potential and mass step, Phys. Rev. A, 59 (1999), 107.
    [8] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, John Wiley and Sons Inc., New York, 1990.
    [9] L. Serra, E. Lipparini, Spin response of unpolarized quantum dots, EPL, 40 (1997), 667.
    [10] P. Goetsch, R. Graham, Linear stochastic wave equations for continuously measured quantum systems, Phys. Rev. A, 50 (1994), 5242.
    [11] R. N. Costa Filho, M. P. Almeida, G. A. Farias, et al. Displacement operator for quantum systems with position-dependent mass, Phys. Rev. A, 84 (2011), 050102.
    [12] B. Gonrul, O. Ozer, B. Gonul, et al. Exact solutions of effective-mass Schrödinger equations, Mod. Phys. Lett. A, 17 (2002), 2453-2465. doi: 10.1142/S0217732302008514
    [13] A. P. Zhang, P. Shi, Y. W. Ling, et al. Solutions of one-dimensional effective mass Schrödinger equation for PT-symmetric Scarf potential, Acta Phys. Pol. A, 120 (2011), 987-991. doi: 10.12693/APhysPolA.120.987
    [14] T. K. Jana, P. Roy, Potential algebra approach to position-dependent mass Schrödinger equations, EPL, 87 (2009), 30003.
    [15] M. Sebawe Abdalla, H. Eleuch, Exact solutions of the position-dependent-effective mass Schrödinger equation, AIP Adv., 6 (2016), 055011.
    [16] G. C. Wu, Z. G. Deng, D. Baleanu, et al. New variable-order fractional chaotic systems for fast image encryption, Chaos, 29 (2019), 083103.
    [17] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
    [18] G. C. Wu, D. Baleanu, W. H. Luo, Lyapunov functions for Riemann-Liouville-like fractional difference equations, Appl. Math. Comput., 314 (2017), 228-236.
    [19] K. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications Inc., Mineola, New York, 2002.
    [20] G. C. Wu, D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dynam., 75 (2014), 283-287. doi: 10.1007/s11071-013-1065-7
    [21] R. Yilmazer, N-fractional calculus operator Nμ method to a modified hydrogen atom equation, Math. Commun., 15 (2010), 489-501.
    [22] R. Yilmazer, M. Inc, F. Tchier, et al. Particular solutions of the confluent hypergeometric differential equation by using the nabla fractional calculus operator, Entropy, 18 (2010), 49.
    [23] R. Yilmazer, O. Ozturk, On nabla discrete fractional calculus operator for a modified Bessel equation, Therm. Sci., 22 (2018), S203-S209.
    [24] O. Ozturk, R. Yilmazer, Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68 (2019), 833-839.
    [25] R. Yilmazer, N. S. Demirel, Discrete fractional solutions of a Gauss equation, AIP Conference Proceedings, 2037 (2018), 020029.
    [26] R. Yilmazer, Discrete fractional solution of a non-homogeneous non-fuchsian differential equations, Therm. Sci., 23 (2019), S121-S127.
    [27] R. Yilmazer, S. Karabulut, Solutions of the generalized Laguerre differential equation by fractional differ integral, AIP Conference Proceedings, 2037 (2018), 020030.
    [28] J. B. Diaz, T. J. Osler, Differences of fractional order, Math. Comput., 28 (1974), 185-202. doi: 10.1090/S0025-5718-1974-0346352-5
    [29] F. M. Atıcı, P. W. Eloe. Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theo., 3 (2009), 1-12.
    [30] N. Acar, F. M. Atıcı, Exponential functions of discrete fractional calculus, Appl. Anal. Discrete Math., 7 (2013), 343-353. doi: 10.2298/AADM130828020A
    [31] F. M. Atıcı, P. W. Eloe, Gronwall's inequality on discrete fractional calculus, Comput. Math. Appl., 64 (2012), 3193-3200. doi: 10.1016/j.camwa.2011.11.029
    [32] C. W. Granger, R. Joyeux, An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal., 1 (1980), 15-29. doi: 10.1111/j.1467-9892.1980.tb00297.x
    [33] J. R. M. Hosking, Fractional differencing, Biometrika, 68 (1981), 165-176. doi: 10.1093/biomet/68.1.165
    [34] H. L. Gray, N. Zhang, On a New definition of the fractional difference, Math. Comput., 50 (1988), 513-529. doi: 10.1090/S0025-5718-1988-0929549-2
    [35] C. Tezcan, R. Sever. O. Yesiltas, A new approach to the exact solutions of the effective mass Schrödinger equation, Int. J. Theor. Phys., 47 (2008), 1713-1721. doi: 10.1007/s10773-007-9613-x
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