Startseite Technik Shehu transform on time-fractional Schrödinger equations – an analytical approach
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Shehu transform on time-fractional Schrödinger equations – an analytical approach

  • Mamta Kapoor ORCID logo EMAIL logo
Veröffentlicht/Copyright: 8. August 2022
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Abstract

In the present study, time-fractional Schrödinger equations are dealt with for the analytical solution using an integral transform named Shehu Transform. Three kinds of time-fractional Schrödinger equations are discussed in the present study. Shehu transform is utilized to reduce the time-fractional PDE along with the fractional derivative in the Caputo sense. The present method is easy to implement in the search for an analytical solution. As no discretization or numerical program is required, the present scheme will surely be helpful in finding the analytical solution to some complex-natured fractional PDEs.


Corresponding author: Mamta Kapoor, Department of Mathematics, Lovely Professional University, Phagwara, Punjab 144411, India, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] K. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, 1974.Suche in Google Scholar

[2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993.Suche in Google Scholar

[3] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications, vol. 1, Science and Technica, 1987.Suche in Google Scholar

[4] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.Suche in Google Scholar

[5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, 2006.Suche in Google Scholar

[6] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, vol. 84, Springer Science & Business Media, 2011.10.1007/978-94-007-0747-4Suche in Google Scholar

[7] S. Das, Functional Fractional Calculus, Springer Science & Business Media, 2011.10.1007/978-3-642-20545-3Suche in Google Scholar

[8] R. Hilfer, Ed. Applications of Fractional Calculus in Physics, World Scientific, 2000.10.1142/3779Suche in Google Scholar

[9] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, vol. 35, New York, Springer, 2003.10.1007/978-0-387-21746-8Suche in Google Scholar

[10] L. Debnath, “Recent applications of fractional calculus to science and engineering,” Int. J. Math. Math. Sci., vol. 2003, no. 54, pp. 3413–3442, 2003. https://doi.org/10.1155/s0161171203301486.Suche in Google Scholar

[11] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010.10.1142/p614Suche in Google Scholar

[12] D. Baleanu, Z. B. Güvenç, and J. T. Machado, Eds. New Trends in Nanotechnology and Fractional Calculus Applications, New York, Springer, 2010, p. C397.10.1007/978-90-481-3293-5Suche in Google Scholar

[13] R. Herrmann, Fractional Calculus: An Introduction for Physicists, 2011.10.1142/8072Suche in Google Scholar

[14] A. Papoulis, “A new method of inversion of the Laplace transform,” Q. Appl. Math., vol. 14, no. 4, pp. 405–414, 1957. https://doi.org/10.1090/qam/82734.Suche in Google Scholar

[15] A. Kılıçman and H. E. Gadain, “On the applications of Laplace and Sumudu transforms,” J. Franklin Inst., vol. 347, no. 5, pp. 848–862, 2010. https://doi.org/10.1016/j.jfranklin.2010.03.008.Suche in Google Scholar

[16] T. M. Elzaki, “On the connections between Laplace and Elzaki transforms,” Adv. Theor. Appl. Math., vol. 6, no. 1, pp. 1–11, 2011.Suche in Google Scholar

[17] M. S. Rawashdeh and S. Maitama, “Solving coupled system of nonlinear PDE’s using the natural decomposition method,” Int. J. Pure Appl. Math., vol. 92, no. 5, pp. 757–776, 2014. https://doi.org/10.12732/ijpam.v92i5.10.Suche in Google Scholar

[18] S. Maitama and W. Zhao, “New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations,” Int. J. Nonlinear Anal. Appl., vol. 17, no. 2, pp. 167–219, 2019.Suche in Google Scholar

[19] D. Ziane, R. Belgacem, and A. Bokhari, “A new modified Adomian decomposition method for nonlinear partial differential equations,” Open Journal of Mathematical Analysis (OMA), vol. 3, no. 2, pp. 81–90, 2019. https://doi.org/10.30538/psrp-oma2019.0041.Suche in Google Scholar

[20] L. Akinyemi and O. S. Iyiola, “Exact and approximate solutions of time‐fractional models arising from physics via Shehu transform,” Math. Methods Appl. Sci., vol. 43, no. 12, pp. 7442–7464, 2020. https://doi.org/10.1002/mma.6484.Suche in Google Scholar

[21a] R. Belgacem, D. Baleanu, and A. Bokhari, Shehu Transform and Applications to Caputo-Fractional Differential Equations, 2019.Suche in Google Scholar

[b] G. H. Hardy, “Gösta Mittag-Leffler, 1846–1927,” Proc. R. Soc. Lond. (A), vol. 119, 1928.Suche in Google Scholar

[22] A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler function and its properties,” J. Math. Anal. Appl., vol. 336, no. 2, pp. 797–811, 2007. https://doi.org/10.1016/j.jmaa.2007.03.018.Suche in Google Scholar

[23] O. S. Iyiola, E. O. Asante-Asamani, and B. A. Wade, “A real distinct poles rational approximation of generalized Mittag-Leffler functions and their inverses: applications to fractional calculus,” J. Comput. Appl. Math., vol. 330, pp. 307–317, 2018. https://doi.org/10.1016/j.cam.2017.08.020.Suche in Google Scholar

[24] Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, 2003.10.1016/B978-012410590-4/50012-7Suche in Google Scholar

[25] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys., vol. 71, no. 3, p. 463, 1999. https://doi.org/10.1103/revmodphys.71.463.Suche in Google Scholar

[26] J. Belmonte-Beitia and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett., vol. 373, no. 4, pp. 448–453, 2009. https://doi.org/10.1016/j.physleta.2008.11.056.Suche in Google Scholar

[27] T. Xu, B. Tian, L. L. Li, X. Lü, and C. Zhang, “Dynamics of Alfvén solitons in inhomogeneous plasmas,” Phys. Plasmas, vol. 15, no. 10, p. 102307, 2008. https://doi.org/10.1063/1.2997340.Suche in Google Scholar

[28] M. Naber, “Time fractional Schrödinger equation,” J. Math. Phys., vol. 45, no. 8, pp. 3339–3352, 2004. https://doi.org/10.1063/1.1769611.Suche in Google Scholar

[29] S. Wang and M. Xu, “Generalized fractional Schrödinger equation with space-time fractional derivatives,” J. Math. Phys., vol. 48, no. 4, p. 043502, 2007. https://doi.org/10.1063/1.2716203.Suche in Google Scholar

[30] S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Phys. Lett., vol. 372, no. 5, pp. 553–558, 2008. https://doi.org/10.1016/j.physleta.2007.06.071.Suche in Google Scholar

[31] R. K. Saxena, R. Saxena, and S. L. Kalla, “Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics,” Appl. Math. Comput., vol. 216, no. 5, pp. 1412–1417, 2010. https://doi.org/10.1016/j.amc.2010.02.041.Suche in Google Scholar

[32] J. Wang, Y. Zhou, and W. Wei, “Fractional Schrödinger equations with potential and optimal controls,” Nonlinear Anal. R. World Appl., vol. 13, no. 6, pp. 2755–2766, 2012. https://doi.org/10.1016/j.nonrwa.2012.04.004.Suche in Google Scholar

[33] N. A. Khan, M. Jamil, and A. Ara, Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method, International Scholarly Research Notices, 2012, 2012.10.5402/2012/197068Suche in Google Scholar

[34] K. M. Hemida, K. A. Gepreel, and M. S. Mohamed, “Analytical approximate solution to the time-space nonlinear partial fractional differential equations,” Int. J. Pure Appl. Math., vol. 78, no. 2, pp. 233–243, 2012.Suche in Google Scholar

[35] S. H. Hamed, E. A. Yousif, and A. I. Arbab, “Analytic and approximate solutions of the space-time fractional Schrödinger equations by homotopy perturbation Sumudu transform method,” in Abstract and Applied Analysis, vol. 2014, Hindawi, 2014.10.1155/2014/863015Suche in Google Scholar

[36] S. O. Edeki, G. O. Akinlabi, and S. A. Adeosun, “Analytic and numerical solutions of time-fractional linear Schrödinger equation,” Commun. Math. Appl., vol. 7, no. 1, pp. 1–10, 2016.Suche in Google Scholar

[37] A. Mohebbi, M. Abbaszadeh, and M. Dehghan, “The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics,” Eng. Anal. Bound. Elem., vol. 37, no. 2, pp. 475–485, 2013. https://doi.org/10.1016/j.enganabound.2012.12.002.Suche in Google Scholar

[38] K. Shah, M. Junaid, and N. Ali, “Extraction of Laplace, Sumudu, Fourier and Mellin transform from the natural transform,” J. Appl. Environ. Biol. Sci, vol. 5, no. 9, pp. 108–115, 2015.Suche in Google Scholar

[39] A. M. Malik and O. H. Mohammed, “Two efficient methods for solving fractional Lane–Emden equations with conformable fractional derivative,” J. Egypt. Math. Soc., vol. 28, no. 1, pp. 1–11, 2020. https://doi.org/10.1186/s42787-020-00099-z.Suche in Google Scholar

[40] S. Ali, S. Bushnaq, K. Shah, and M. Arif, “Numerical treatment of fractional order Cauchy reaction diffusion equations,” Chaos, Solit. Fractals, vol. 103, pp. 578–587, 2017. https://doi.org/10.1016/j.chaos.2017.07.016.Suche in Google Scholar

[41] K. Shah, H. Naz, M. Sarwar, and T. Abdeljawad, “On spectral numerical method for variable-order partial differential equations,” AIMS Math., vol. 7, no. 6, pp. 10422–10438, 2022. https://doi.org/10.3934/math.2022581.Suche in Google Scholar

[42] A. Bashan, N. M. Yagmurlu, Y. Ucar, and A. Esen, “An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method,” Chaos, Solit. Fractals, vol. 100, pp. 45–56, 2017. https://doi.org/10.1016/j.chaos.2017.04.038.Suche in Google Scholar


Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/ijnsns-2021-0423).


Received: 2021-11-06
Revised: 2022-06-01
Accepted: 2022-06-19
Published Online: 2022-08-08

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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