Spectral structure and solution of fractional hydrogen atom difference equations

Erdal Bas; Ramazan Ozarslan; Resat Yilmazer; Erdal Bas; Ramazan Ozarslan; Resat Yilmazer

doi:10.3934/math.2020093

AIMS Mathematics

2020, Volume 5, Issue 2: 1359-1371. doi: 10.3934/math.2020093

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Spectral structure and solution of fractional hydrogen atom difference equations

Department of Mathematics, Science Faculty, Firat University, 23119 Elazig, Turkey

Received: 30 August 2019 Accepted: 08 January 2020 Published: 20 January 2020
MSC : 34B24, 39A70, 34A08

In this study, discrete fractional hydrogen atom (DFHA) operators are presented. Hydrogen atom differential equations have series solution due to having singularity and also obtaining a series solution for DFHA equations have some difficulties, for this reason, we study to obtain solution of DFHA equations by means of nabla Laplace transform. In addition to all these, we show self-adjointness of the DFHA operator and some spectral properties, like orthogonality of distinct eigenfunctions, reality of eigenvalues. Finally, we find an analytical solution of the problem under different q (t) potential functions, different fractional orders and different eigenvalues and the results obtained are illustrated by tables and simulations.
- hydrogen atom,
- discrete fractional,
- electron,
- eigenvalue,
- Schrödinger equation,
- energy level
Citation: Erdal Bas, Ramazan Ozarslan, Resat Yilmazer. Spectral structure and solution of fractional hydrogen atom difference equations[J]. AIMS Mathematics, 2020, 5(2): 1359-1371. doi: 10.3934/math.2020093

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Abstract

In this study, discrete fractional hydrogen atom (DFHA) operators are presented. Hydrogen atom differential equations have series solution due to having singularity and also obtaining a series solution for DFHA equations have some difficulties, for this reason, we study to obtain solution of DFHA equations by means of nabla Laplace transform. In addition to all these, we show self-adjointness of the DFHA operator and some spectral properties, like orthogonality of distinct eigenfunctions, reality of eigenvalues. Finally, we find an analytical solution of the problem under different q (t) potential functions, different fractional orders and different eigenvalues and the results obtained are illustrated by tables and simulations.

References

[1]	J. B. Diaz, T. J. Osler, Differences of fractional order, Math. Comput., 28 (1974), 185-202. doi: 10.1090/S0025-5718-1974-0346352-5
[2]	K. S. Miller, B. Ross, Fractional difference calculus, Proceedings of the international symposium on univalent functions, fractional calculus and their applications, 1988.
[3]	H. L. Gray, N. F. Zhang, On a new definition of the fractional difference, Math. Comput., 50 (1988), 513-529. doi: 10.1090/S0025-5718-1988-0929549-2
[4]	C. Goodrich, A. C. Peterson, Discrete fractional calculus, Berlin: Springer, 2015.
[5]	D. Baleanu, S. Rezapour, S. Salehi, On some self-adjoint fractional finite difference equations, J. Comput. Anal. Appl., 19 (2015).
[6]	G. C. Wu, D. Baleanu, S. D. Zeng, et al. Mittag-Leffler function for discrete fractional modelling, J. King Saud Univ. Sci., 28 (2016), 99-102. doi: 10.1016/j.jksus.2015.06.004
[7]	K. Ahrendt, T. Rolling, L. Dewolf, et al. Initial and boundary value problems for the caputo fractional self-adjoint difference equations, EPAM, 2 (2016), 105-141.
[8]	F. M. Atıcı, P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theo., 2009 (2009), 1-12.
[9]	F. Atici, P. Eloe, Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, 137 (2009), 981-989.
[10]	G. A. Anastassiou, Right nabla discrete fractional calculus, Int. J. Difference Equ., 6 (2011), 91-104.
[11]	T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. EquNY, 2013 (2013), 36.
[12]	T. Abdeljawad, D. Baleanu, Fractional Differences and Integration by Parts, J. Comput. Anal. Appl., 13 (2011).
[13]	T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611. doi: 10.1016/j.camwa.2011.03.036
[14]	T. Abdeljawad, F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), 1-13.
[15]	T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013.
[16]	T. Abdeljawad, R. Mert, A. Peterson, Sturm Liouville Equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaestiones Mathematicae, 49 (2019), 1271-1289.
[17]	J. Hein, Z. McCarthy, N. Gaswick, et al. Laplace transforms for the nabla-difference operator, Pan American Mathematical Journal, 21 (2011), 79-96.
[18]	J. F. Cheng, Y. M. Chu, Fractional difference equations with real variable, Abstr. Appl. Anal., 2012 (2012).
[19]	D. Mozyrska, M. Wyrwas, Solutions of fractional linear difference systems with Caputo-type operator via transform method, ICFDA'14 International Conference on Fractional Differentiation and Its Applications, 2014 (2014), 1-6.
[20]	L. Abadias, C. Lizama, P. J. Miana, et al. On well-posedness of vector-valued fractional differential-difference equations, Discrete Cont. DYN-A, 39 (2019), 2679-2708. doi: 10.3934/dcds.2019112
[21]	I. K. Dassios, A practical formula of solutions for a family of linear non-autonomous fractional nabla difference equations, J. Comput. Appl. Math., 339 (2018), 317-328. doi: 10.1016/j.cam.2017.09.030
[22]	J. Prakash, M. Kothandapani, V. Bharathi, Numerical approximations of nonlinear fractional differential difference equations by using modified He-Laplace method, Alex. Eng. J., 55 (2016), 645-651. doi: 10.1016/j.aej.2015.12.006
[23]	C. N. Angstmann, B. I. Henry, Generalized fractional power series solutions for fractional differential equations, Appl. Math. Lett., 102 (2020), 106107.
[24]	R. Saleh, M. Kassem, S. M. Mabrouk, Exact solutions of nonlinear fractional order partial differential equations via singular manifold method, Chinese J. Phys., 61 (2019), 290-300. doi: 10.1016/j.cjph.2019.09.005
[25]	B. Ahmad, M. Alghanmi, S. K. Ntouyas, et al. A study of fractional differential equations and inclusions involving generalized Caputo-type derivative equipped with generalized fractional integral boundary conditions, 2018.
[26]	E. Bas, F. Metin, Fractional singular Sturm-Liouville operator for Coulomb potential, Adv. Differ. Equ-NY, 2013 (2013), 300.
[27]	E. Bas, The Inverse Nodal problem for the fractional diffusion equation, Acta Sci-Technol, 37 (2015), 251-257. doi: 10.4025/actascitechnol.v37i2.17273
[28]	M. Klimek, Om P. Agrawal, On a regular fractional Sturm-Liouville problem with derivatives of order in (0, 1), Proceedings of the 13th International Carpathian Control Conference (ICCC), 2012.
[29]	M. Dehghan, A. B. Mingarelli, Fractional Sturm-Liouville eigenvalue problems, I, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 114 (2020), 1-15.
[30]	E. S. Panakhov, A. Ercan, Stabılıty Problem Of Sıngular Sturm-Lıouvılle Equatıon, Twms Journal Of Pure And Applıed Mathematıcs, 8 (2017), 148-159.
[31]	T. Gulsen, An inverse nodal problem for p-Laplacian Sturm-Liouville equation with Coulomb potential, J. Nonlinear Sci. Appl., 10 (2017), 5393-5401. doi: 10.22436/jnsa.010.10.24
[32]	E. Bas, R. Ozarslan, Sturm-Liouville Problem via Coulomb Type in Difference Equations, Filomat, 31 (2017), 989-998. doi: 10.2298/FIL1704989B
[33]	E. Bas, R. Ozarslan, Sturm-Liouville Difference Equations Having Special Potentials, Journal of Advanced, 6 (2017), 529-533.
[34]	R. Almeida, A. Malinowska, M. Morgado, et al. Variational methods for the solution of fractional discrete/continuous Sturm-Liouville problems, J. Mech. Mater. Struct., 12 (2017), 3-21. doi: 10.2140/jomms.2017.12.3
[35]	E. Bas, R. Ozarslan, Theory of discrete fractional Sturm-Liouville equations and visual results, AIMS Mathematics, 4 (2019), 576-595. doi: 10.3934/math.2019.3.576
[36]	M. Bohner, T. Cuchta, The Bessel difference equation, Proceedings of the American Mathematical Society, 145 (2017), 1567-1580.
[37]	M. Bohner, T. Cuchta, The generalized hypergeometric difference equation, Demonstratio Mathematica, 51 (2018), 62-75. doi: 10.1515/dema-2018-0007
[38]	T. J. Cuchta, Discrete analogues of some classical special functions, 2015.
[39]	B. M. Levitan, I. S. Sargsian, Introduction to spectral theory: selfadjoint ordinary differential operators: Selfadjoint Ordinary Differential Operators, American Mathematical Soc., 1975.
[40]	E. S. Panakhov, R. Yilmazer, A Hochstadt-Lieberman theorem for the hydrogen atom equation, Appl. Comput. Math., 11 (2012), 74-80.
[41]	E. Bas, E. Panakhov, R. Yılmazer, The uniqueness theorem for hydrogen atom equation, TWMS Journal of Pure and Applied Mathematics, 4 (2013), 20-28.
[42]	M. Bohner, A. C. Peterson, Advances in dynamic equations on time scales, Springer Science & Business Media, 2002.

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